The details of how to perform Shiva Linga Archana(including the details of mantras to be used etc) are briefed in the Lingaarchana chapter of LingaPurana:
As Rishis asked Lomaharshana Maharshi as to the Procedure of
Lingaarchana or Worship the Shiva Linga, the Maharshi stated that once
Devi Parvati also inquired of the same and Shiva Himself revealed the
Procedure:
And, we gather the following information regarding what are the objects/items that should be offered to a Shiva Linga:
Maharshi Lomavarshana now described the actual Procedure of
Rudraabhisheka as follows......Then Shiva is seated at the Center
firmly; He would be seated as Sadyojata Shiva with the Mantra of
Vaamaa Deva; Rudra Gayatri is seated with Aghora Mantra; and Ishana is
seated by reciting
Ishaanassarva Vidyaanaam Ishwarassarva Bhutaanaam
Brahmaadipatih Brahmanoddhi Patih Brahma Shivomey astu Sadaa Shivom/
Then the devotee should worship Shiva Linga with
Paadya-Arghya-Achamanaas; perform Snaana with Gandha-Chandana;
implement formal Snaana with Pancha-Gavya or of Cow’s Five Bye
Products; carry out Abhishka with Ghee, Honey, Sugar, Pavitra Jala
along Pranava Naada. There after, clean up the residues of the
Abhisheka material on the Linga with a clean cloth and decorate the
Linga with flower garlands comprising fresh flowers like Jaati,
Champaka, Kapura, Kannera, Chameli, and Kadamba
So,for abhisekha apart from water(including water from holy rivers like Ganges), the Pancha Gavyas & the Panchamritas are required alongwith sandal paste.
Further info gathered from this chapter of the same Purana:
In all the months, the puja must be performed by lotus either in the
actual form preferably or by way of gold or silver or copper or any
other forms. Puja by Bilva Patra (Shri Patra) again in actual form or
of replicas would be a ‘must’. Devi Lakshmi is stated to reside in bel
patras, while Neel Kamal is the resident of Ambika, Shanmukha stays in
Utpala Pushpa and Maha Deva is fond of Padma. Dhupa or smoke made of
‘Guggula’ and Deepa are integral parts of the Puja as they are the
demolishers of all kinds of sins.
For naivedya Lord Shiva likes Anna(rice) as stated below:
As for Naivedya, ‘Shuddhhaanna’ would satisfy Maha Deva; in fact,
‘Anna Daana’ is one significant charity to the needy that Shiva is
always desirous of on any day, especially on Puranamasi.
Ingredients for Shiva Abhisheka on special months are also given in the same chapter like:
Kartika Month: The entire month is of significance and daily
Shivaarchana-Abhisheka with Sacred water, Milk, Coconut water, Curd,
Ghee, Chandana, and Bhasma with Maha Nyasa Purvaka, Maha Rudra Purvaka
Snaanaas to Shiva Linga,
Relative importance of offering/bathing Lingam with water,milk,ghee etc , taken from this chapter of the Linga Purana, are as follows:
Mere ‘darshana’ of such Sacred Tirthas is of considerable value, but
‘sparshana’ or touch and fondling of a Shiva Linga is hundred times
better; Abhisheka of Shiva Lingas with water is hundred times better;
Abhisheka with milk is thousand times better; far better is the
Abhisheka with curds; with Madhu or honey it is regarded as better
still and even better would be the Abhisheka by ghee.
I am unable to find,though, the exact rewards for offering each of the above items to Shiva Lingam.Although I think that they are present in some Scriptures(telling you that from my memory). Will be on look out for those.
EDIT-In general by the worship of Shiva Linga one gets everything starting from worldly benefits to Moksha,but I'm yet found the exact fruits of offering specific objects.
From the Shiva Linga Pratishta Vidhi and Mahatmya chapter of Linga Purana:
Linga Devi Uma Devi Lingam Saakshaanmaheswarah /
Tayoh sapujanaa Deva Devascha Pujitou,
Pratishthayaa cha Devesho Devyaa saardham prtishthitah/
Muley Brahmaa vasati Bhagavan Madhya bhaagey cha
Vishnuh Sarveshaanah Pashupati rajo Rudramurthirvarenyah,
Tasmaalingam Gurutatataram Pujatsthaapayedhaa
yasmaad pujyo Ganapatirasau Deva Mukhyaih samastaih/
Gandhaih snugdhupa deepaih snapanahuta balistotrra
mantropahaaraih nityam yebhyarchayaanti tridasha vastunam Linga
murtey Mahesham/
Tasmaat Bhaktopchaarena Staapayet Parameshwaram,
Pujayeccha visheshena Lingam Sarvaartha siddhayai/
Meaning-
The Vedi of the Shiva Linga is Uma Devi herself and the Linga is
Maheswara himself; thus the Linga ought to be installed along with the
Vedi. The Shiva Linga’s bottom is signified as Brahma and the middle
part is signified as Vishnu and the top portion is Maheswara himself.
This is why the high significance of the Rudra Swarupa Pashupati. By
installing the Linga thus, regular pujas should be performed most
sincerely along with Gandha, Dhupa, Deepa, Snaana, Havana, Bali,
Stotra, Mantra and Upahaara. Nitya Puja is a must and this would
ensure freedom from ‘Punarjanma’ or rebirth, Shiva Loka Prapti and
worship-worthiness of Deva-Gandharva-Siddha and Shiva ganaas.