There are in fact more than one question in your question viz:
1.How many types of Siddhis are there?
2)How to achieve them?
3)What is Mantra Siddhi & how to achieve it?
How many types of Siddhis are there?
The Linga Purana describes the primary Siddhis as follows
A successful Yoga Practitioner is endowed with ‘Dasha Siddhis’viz.
Pratibha, Shravana, Vaarta, Darshana, Aaswaada and Vedana. Jnaana
Pratibha is the Siddhi of knowing about a Vastu or Vyavahar or of a
thing of past, present or of future; this Siddhi would infuence or
impress others. Shravana Siddhi enables a Yogi to hear or understand
any kind of sound or conversation or a happening irrespective of
distance or time. Vaartaa Siddhi facilitates the experience of Pancha
Tanmatraas of sabda-sparsha-rupa-rasa and gandhas. Darshana Siddhi is
the ability to perceive or vision of any thing irrespective of time
and distance limitations. Aaswaada Siddhi enables tasting any
substance and Vedana Siddhi is the power of sparsha to experience the
shape, form or a feature of a person or thing.
More kinds of Siddhis are also described on the same page as follows:
In fact a Maha Yogi is stated to possess sixty four kinds of
Paisachika, Parthiva, Raakshasa, Yaaksha, Gaandharva, Aindra,
Vyomatmika, Praajaapatya, Brahmaadi Siddhis but he should discard all
such powers in the quest of Shivatwa. Such Siddhis range from assuming
any type of Swarupa of fatness, slimnes, childhood, youth, old age,
man, woman, any specie of a bird-animal-reptile, mountain, water body
and so on; ability to lift mountains, drink up an ocean, flying on
sky, passing through a needle’s eye and endless such miracles. [The
normal Siddhis are Anima (miniaturising), Mahima ( Maximising),
Prakamya (Visioning and Hearing), Isatwa (Rulership), Vasitwa (Self
Control), Kama Vasitwa (ability to fulfill desires), Doora Shravana
and Doora Darshana or distant vision and hearing, Parakaya Pravesha
or ability to enter other’s bodies; ‘Devaanaam Saha kreedanam’ or
playing with Devas, Yatha Sankalpa Samsiddhi or instant fulfillment of
desires,Triloka Jnaana or knowledge of the happenings of Three Lokas;
Control of heat and cold, Paraajaya or invincibility and so on.]
How to achieve Siddhis?
The method of obtaining Siddhis is by practising "Ashtanga Yoga".The description of Ashtanga Yoga is found on the same page from where i quote:
To control the activities or goings on in the mind is Yoga. The
Ashtanga or Eight-Limbed Yoga comprises of Yama- Niyama- Aasana-
Pranaayama- Pratyahara-Dharana-Dhyana-Samadhi. Yama is denoted by good
behaviour comprising Ahimsa (Non-Violence), Satya (Truth),
Asteya (stealing), Brahmacharya (avoidance of sex) and Aparigraha or
Excessive acquisition. Basic Niyamas is in fact an extension of Yama:
To refrain from hurting or torturing co-Beings physically or mentally
is Ahimsa; to convey whatever is seen, heard, and believed is Satya
unless it does not hurt any body is known as Satya; to desist stealing
of other’s property by deed, thought or otherwise is called Asteya; to
negate from relationship with another female or in respect of a female
wih another male, excepting the wedded woman or man as the case may
be and this should be observed by vision, thought or deed is called
Brahmacharya or celibacy; and to abstain from excessive acquisition of
materials far and above one’s needs in the short run of time is stated
to be Aparigraha. Besides the above General Principles, Niyamas also
involve the specific deeds viz. Shoucha or Physical
Cleanliness,Yagna-Tapa or meditation, Daana or charity, Swadhyaya or
Memorising or Reciting the Scriptures, Indriya Nigraha or Control of
Physical Limbs, Vratas like Chandrayana as also or to perform formal
worship directed to specific Devatas and purposes,,
What is Mantra Siddhi & how to achieve it?
Mantra Siddhi is a devotee's full control over the wholesome power of a Mantra & is obtained upon reciting a particular Mantra for a fixed number of times with certain other constraints being imposed .First and foremost you should get your Guru to obtain a Mantra.Or else you can pray to your Ista Devta and start.Rest is quoted from this page.
Mantra Siddhi is also called Mantra Purasarchana.
METHOD OF MANTRA PURASCHARANA
The repetition of a Mantra with rigid spiritual observances a fixed
number of times to obtain quick spiritual progress is known as Mantra
Purascharana. It can be performed for material progress too. The
practitioner should observe certain rules and undergo strict dietetic
discipline to ensure quick Mantra Siddhi.
During the Purascharana take only fresh vegetables, fruits, milk,
roots, barley and Havis-Anna (rice cooked with ghee, sugar, milk). A
Sadhaka can live on pure Bhiksha (alms) also. If you can live on milk
alone during the period of Purascharana it is highly laudable. You can
have Mantra Siddhi even by repeating the Mantra a lakh of times.
Select any holy place of pilgrimage on the banks of sacred Ganga,
confluence of rivers, mountain valleys of charming scenery, temples,
Tulasi gardens, below Asvattha trees or convert a portion of your
house into a temple by keeping the picture of the Lord, burning
incense etc., and by suitable decorations. Purascharana done in holy
places has a benefit hundred times superior to that done in one's own
You can select any Mantra for Purascharana. Your Guru Mantra or Ishta
Mantra is the best. Sandhya time, sunrise, sunset, midday are all
recommended for Japa. Repeat the Mantra as many lakhs of times as
there are letters in the Mantra. You can do half of that number. In no
case the number should be less than a lakh........
After completing the Purascharana perform Homa th the number of Japa,
Tarpana (water libations) th the number of Homa, Marjana (sprinkling)
th the number of Tarpana and feeding of Brahmins th the number of
Tarpana. You can do feeding and charity according to your capacity if
you cannot adhere to the above strictly.
It is recommended that you read the whole article and not just the portions i have quoted from it.