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In the book that I read recently (Mantram Handbook by Eknath Easwaran), he advises that one should repeat the mantram silently, whenever the mind is free, in day to day instances also, like waiting in a queue, etc. Overall, the sense of his conception of mantra seems to be limited to repeating some word along with an associated idea. This seems like some elementary meditation practice of keeping the mind fixed and not letting it wander, something like the popular, Buddhist Mindfulness meditation. Most of his quotations are either Gandhi, or Christians like some Brother Lawrence, Teresa, Meister Ekhart but also refer to a few sayings of Sri Ramakrishna, that too for namesake. Given that this is the case, I don't know how reliable his literature is on this subject. (His commentary on Upanishads-Essence of the Upanishads seemed pretty good)
However, as far as I have heard, mantra has an associated sound/vibration component too. Its not just repeating of words and ideas. Also, repeating mantras is said to be part of the integrated Raja Yoga/Mantra Yoga systems, and I don't think it is effective if one simply picks and chooses components that suit their religious sensibilities. It is regarded as being dangerous too. For example, Swami Vivekananda, in one of his lectures (I think raja-yoga), has said that Mohammad, the founder of the Islamic religion was a misguided person, because he didn't properly understand what he was doing(raja yoga according to Swamiji) and it led him to founding such a violent creed that preaches intolerance of any criticism.
Edit 1: Recently, I found this link, which advocates chanting Krishna's name while working, standing, etc. everywhere. The transcript is a bit long to read, here's the relevant part in short:

Śyāmasundara: You were saying earlier that we can also supplement our Kṛṣṇa consciousness while we're working, hammering the nails.

Prabhupāda: Yes.

Śyāmasundara: So chanting along with devotional service, performing our duties while concentrating on Kṛṣṇa, is also part of the process, isn't it?

Prabhupāda: Yes. Anything, any way. The whole idea is manāḥ kṛṣṇe niveṣayet (SB 7.1.32). Mind should be fixed up in Kṛṣṇa. That is the process. Either you go through philosophy or through arguments or through chanting. Any way. That is recommended in the Bhagavad-gītā. Yoginām api sarveṣāṁ (BG 6.47). Of all kinds of yogīs. In the . . . you might have read it. Yoginām api sarveṣāṁ. I think Maharshi has translated this Bhagavad-gītā? And in the Sixth Chapter . . .? You have read it?

Easwaran, in his book, says that the rhythm in which it is repeated does not matter, whether it is repeated fast or slow does not matter. Again, I find this odd, as I have seen people associated with the tradition observing various rules while chanting mantras aloud. Also, I happened to glance over some of the pages of Mantra Yoga by Swami Sivananda Saraswati. I don't think Easwaran's views are supported there either. In this context, my question can be divided into the following sub-parts:

  1. What are the various uses of mantras? By this I also mean, Can they be used for non-visual meditation too? What do the various Hindu spiritual leaders, acharyas, saints and the scriptures say on these various uses?
  2. What are some good resources/books to learn about mantras on your own for Hindus. I mention Hindus, because other religions/atheists tend to purposefully skip the contents which stand at odds with their ideologies. I don't have any such restrictions, hence I would like to explore the subject in its entirety. Also, is the mantram practice suitable for someone, who hasn't had a proper initiation by a guru/sanyasin? (I had my upanayana ceremony when I reached of age, but I was too young and rebel minded then and grumpily accepted whatever I was told to do. I considered myself as an atheist at that point of time, due to media/pop culture conditioning. I don't think that counts as initiation.) If so, why? What purpose does initiation serve? Is it only about giving the mantra or also sustained support of an understanding, mature person throughout the spiritual proccess?
  3. Is initiation by a guru absolutely necessary for all mantras? What is the position of this proposition for different mantras, like generic popular mantras used by the Hare Krishna ISKON movement like hare krishna mantra, repeating the name rama-rama , mantras popularly used in almost every hindu household like Om namah Shivay, mantras said to have powerful psychological effects like mantras from the Durga Saptashati (hymns to the goddess).

On reading some links on the answers provided below, I understand that the answerers say that mantras cannot be repeated without initiation, quoting the scriptures directly. However, My question is not limited to the shastras. I would like to understand the position of this statement expressed by the new Hindu gurus also. By new Hindu gurus, I mean people who wrote/practiced Hinduism and preached about it. They might have taken a liberal interpretation of the sastras, not literal meanings in Sanskrit, and they are widely considered Hindu by the large, practicing Hindu community in India and elsewhere and are much held in much esteem by the community. Some examples are Swami Vivekananda, Sri Aurobindo, Sri Ramakrishna, Swami Sivananda Saraswati, Yogananda Paramhansa, Srila Prabhupada, Eknath Easwaran, Sadhguru.
Anyways, the sastras were supposed to be updated with time by men well read in sastras and possessing valid experiences (on the basis of adhikara), and they couldn't be updated traditionally, because the institutions, mathas, and temples were destroyed by invaders and thanks to the secular government, they don't function anymore/ lie defunct. So, in my opinion, maybe, the initiation part and some of the rituals might be outdated. For example, I think Chaitanya Mahaprabhu at the time of Islamic invasions, made some changes too to adjust according to changing times.
As for other uses of mantra, I read in Sundara - Yogic Therapy (primarily a book on hatha yoga), that mantras can be used for pranic therapy and divine healing. He was primarily a hatha yoga guru, but was initiated into vedanta too by a advaita vedantin, so he is also part of the continuous guru-shishya parampara.
Edit2:I remembered another innovative use that I saw a while ago. This video shows British kids reciting Sanskrit shlokas from the Upanishads to improve their pronounciation. Though not technically a mantras, positive thoughts in the Sanskrit language are associated with mantras.I am sure there are other uses as well.
On the last part, about sources, I am not looking for a collection of mantras as such, like the tantras. I wish to read a book, explaining the uses of mantras, the procedure to be followed while reciting a mantra, the various precautions to be taken, about health benefits, too, if any such exist. Please note that when I am asking for such resources, it is not simply for fun/research. I intend to practice. For examples of some such books, Mantra Yoga by Swami Sivananda Saraswati, Mantra Yoga and Primal sound by David Frawley, Meditation and Mantras by Swami Vishnu Devananda.
Another note: They need not be purely Hindu in nature. For example, Vajrayana Buddhism shares many features with Hinduism/Tantra, hence a suitable book can be mentioned too. For Example).
I found these books while surfing internet and goodreads book recommendations. I am sure experts here can lead me to better sources/ verify these are good sources. Please note: I am looking for English sources. Though I am literate in Hindi, I am not comfortable in the language, as I have a habit of reading books in English. Please post Hindi books only if no English alternatives are available/ the Hindi books are much better in quality/calibre than the English ones.

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My human Guru, Swami Swahananda, has written a book on Mantra. You might find the book, 'Meditation and Other Spiritual Disciplines' useful.

I am posting below an excerpt of the book.

Swami Swahananda in his book writes:

Among the Vedic sacred formulas, the Gayatri has been prescribed for repetition from ancient times. The Vedic people had great faith in the efficacy of mantras. The Mimamsakas considered mantras to be the embodiments of deities. In fact, they accepted no deities other than the mantras at all. To them Brahman is nothing but sound (shabda), and sound produces form. Thus, the name embodied in the mantra is more real than the form of the deity. .... Forms are made up of nothing but fine vibrations, and vibrations are produced by sound only, so these ancient conceptions are tenable. From another point of view, a mantra gains spiritual potency through many years of being associated with holy men who have repeated it and attained realization through it. The concept of the mantra is based on the psychological fact that much of our thinking depends on auditory symbols. Repetition of the mantra creates a chain of thought which induces the mind with the thought of God, and this is the aim of all spiritual practices. ................. Every Vedic mantra has a rishi who first intuited it, a particular metre in which it is composed, and a deity to whom it is addressed. In Tantric mantras the most important element is the vija, a sacred syllable considered to be charged with spiritual potency, as also Shakti or power and a kilaka, a pillar (inner syllables on which the mantra rests.) All the letters of the Sanskrit alphabets are considered as different matrikas, minor deities around the main deity. The mantra is charged with a special potency, so much so that the Tantrikas believe that when offerings are made to the deity with the appropriate mantras, the deity accepts them immediately. ......... The Chandogya Upanishad I.1.10 says about the efficacy of the repetition of Om: Both perform spiritual practices - he who knows and he who does not know. But knowledge and ignorance are different in their effects. Whatever is performed with knowledge, faith and meditation, becomes spiritually effective.

Meditation and other spiritual disciplines by Swami Swahananda

Another book which does discuss the topics you want to know is Kularnava Tantra Introduction by Arthur Avalon Readings by M. P. Pandit.

Innumerable are the Mantras; they but distract the mind. Only that Mantra which is received through the Grace of the Guru gives all fulfilment.

Kularnava Tantra, Purascharana, Readings by M.P. Pandit

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Since the question is too broad to cover in detail, we will sort of summarize.

What are various uses of mantra?....

"मननात्‌ त्रायेत यस्मत्तस्मान्मन्त्रः प्रकीर्त्तित:" -Shardatilak Tantra (preface)

From 'm-kara' the 'Mana' from 'tra-kara' the 'rakshan' or protection, which means the thoughts which help us complete (siddhi) the work (karya) are mantras. I.e. The mantras are those lead to Karya siddhi (completion of purpose or work).

There are mantras for every kind of work, according to the type and purpose of the work. In Tantric aspects, the 'ShatKarma' play a very important roll. I dont know what is non-visual meditation.

What are good sources...

The knowledge about mantras is spread across texts. No single text can give a full picture of what's going on. I am not going state any name in specific. The mantras are discussed along with sadhana. You cannot tell what you will find in a certain book or a tantra. See: https://hinduism.stackexchange.com/a/43089/21353

Intiation...

दीयते ज्ञान सदभावः क्षीयते पशुवासना । दान क्षपण संयुक्ता दीक्षा तेनेह कीर्तिता ॥

Providing the Harmony along with ability to reduce the desire is diksha. Intiation is a step towards liberation. (See: What is Dīkṣā (दीक्षा)?).

It depends on the mantra you practice. Many people take Guru Mantra and only chant that, nothing more, nothing less. Some 'mantras' for a perticular purpose mandatorily require intiation. And many 'sadhana' at least a Guru Agnya (permission).

The Generic popular mantras are mostly 'Nama'. It actually does not mean we can freely chant them too. There are instances where we need to only chant name like 'shri ram, jay ram, jay jay ram' placing a 'om' before it changes its property and it no longer remains 'nama'. As I said many mantras can be practiced without intiation but it's true potential can only be achived with intiation. Its better to practice them, instead blaming the intiation part. At least all 'satvik' mantras can be practiced. Reciting stotra is also beneficial, the 'fal-shruti' (what fruits?!) Is all about its benifits.

See: 1 , 2 And links therein.

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  • One should look at the picture of the deity associated with the mantra carefully, close the eyes and visualize the deity in his mind. Try to visualize the whole diety from head to toe. Initially, you will find that you forget the features of the diety pretty soon. Then, open the eyes, and again look at the picture carefully. Keep repeating this for a fixed number of times every day. Later, you will be able to visualize the deity continuously without opening your eyes. Then, later, he proceeds to say that mansik puja is superior to physical puja also.- extracts from Japa Yoga - Swami Sivananda
    – user22892
    Feb 5 at 3:52
  • This is what I refer to as visual meditation. Non-visual meditation is when nothing is visualized, like the Buddhist mindfulness meditation.
    – user22892
    Feb 5 at 3:53
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    Yes, I read that. If I find something I will update the answer and let you know.
    – Proxy
    Feb 5 at 3:59

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