The best way when a snake bites is to run to the hospital for treatment. It is because we are in Kali Yuga now and we rarely find person who have mantra siddhis. We can also rarely do mantra siddhis on our own without Guru.
But we can use mantras on the way while running to hospital or when reaching hospital is not possible.
The fear we have after snakebite is fear of death hence we can chant death removing/ Mahamrityunjaya Mantra for this purpose (while in the way to hospital):
ॐ त्र्यम्बकं यजामहे सुगन्धिं पुष्टिवर्धनम्
उर्वारुकमिव बन्धनान्मृत्योर्मुक्षीय माऽमृतात् ।।
Om tryambakaṃ yajāmahe sugandhiṃ pushtivardhanam
urvārukamiva bandhanānmṛmrityor mokshiye maamritat ।।
Several mantras are given in Panchanga Calendars also. I don't know from where they are taken. It is written in Panchanga calendars that chanting this mantra and blowing air to body will remove poison (might be tested in emergency condition):
ॐ कुरकुलये हुं फट् स्वाहा ।
Om Kurkulaye Hum Phat Swaha
Also Panchanga calendars give some preventive measures to remove fear of snakes before being bitten from certain area/house. They are:
ॐ आस्तिकाय नमः ।
Om Aastikaye Namaha
The above mantra invoke the snake 'Astika' who is saver of snake race in serpent Yagna conducted by Janamjaye in Mahabharat. But I don't know the origin of above mantras. As I have already told they are found in Panchanga calendars. But there is also no loss in chanting these mantras while walking in grassy areas and snake fearing areas as it is of preventive nature ie. Preventing snake bite.
If you are interested in Puranic Methods then Garuda Purana and Agni Purana contains procedures of snake poison removal. It is also told a person who have read or listened or donated Garuda Purana is safe from being bitten by snake as he gains Garuda-tatwa from it.
Anyway Garuda Purana in chapter 20 gives the method for poison removal of snake:
Chapter XX - Mantra-cures (curative formulas) of snakebite as narrated by Śhiva
Suta said:—I will now now describe the highly secret mantrams narrated by Shiva. His weapons are the noose, the bow, the discus, the club, the dart and Pattisha. Having used these weapons inspired with mantrams in a battle a king conquers his enemies. The mantram for purifying the mantrans should be first written on a lotus petal. Om is the Brahma Vijam. Hrim is the Vishnu Vijam. These three Vijams should be assigned to the head of Shiva thrice in order.
Om, Hrim, Hrim.
Having taken up the dart in his hand he should whirl it in the sky. By seeing it all the evil stars and serpents are destroyed. Having held the smoky coloured bow by the hand a man should meditate on it in the sky. By it the wicked serpents, the evil stars, clouds and Rakshasas are destroyed. This mantram protects the three worlds, what to speak of the land of mortals?
Om, jum, Sam, Hum, Phat. Eight sticks of Catechu wood, inspired with mantrams, should be placed on the ground. That will prevent the falling of thunder-bolt. The eight sticks should be inspired with great mantram described by Garuda. The ground should be dug twenty one times in the night. This will ward off the dangers proceeding from lightning, mouse, and thunder-bolt.
The mantram is:—Hara, Kshara, amala, Vashat, added with Vindu Sadasiva.
Om, Hram, salutation unto Sadashiva.
He should then assign pinda (balls of rice) effulgent like Darimi flowers with the fore-finger. By seeing it the evil clouds, lightning and other enemies viz. the Rakshasas, goblins and female ghosts fly away into the ten quarters.
Om, Hrim, salutation unto Ganesha. Om, Hrim, salutation unto the chakra of Sthambhana.
Om, em, salutation unto the Damaras of the three worlds.
This pinda is called Bhairava which counteracts the effect of poison and the evil effects of the hostile planets. It protects the field and grinds the goblins and Rakshasas.
Om, Namas. Having meditated on the thunder-bolt of his hand he should ward off the evil influence of wicked clouds and with Vajra Mudra all the ghosts, the enemies with poison. Om, Kshum, Namas. He should meditate on his left hand. It destroys all venomous creatures.
Om, Hram, Namas. The very recitation of this mantram destroys the evil clouds and stars. Having meditated on death he should consume the universe with the destroying weapon.
Om, Kshma, Namas.
Meditating on Bhairava one should remove the evil influences of stars, goblins and poison.
Om lasat, jhvaksha Svāhā. This mantram destroys the enemies of the field vis, the evil stars, goblins, poison and birds.
Om Ksham Namas. The figure of a cistern should be drawn with blood and then the names of planets should be written there.
Om, Mara, Mara, Maraya, Maraya Svāhā. Om, Hum, Phat, Svāhā.
The dart should be inspired mentally with eight hundred mantrams. It destroys all the enemies.
With higher energies the lower ones should be suppressed. Then the mantrams should be practised in Puraka and should be again well inspired in Kumbhaka. They should then be received with Pranava. When the mantrams are thus properly received and used they yield fruts like servants.
There are also other methods like inserting peepal branch in both ears and other and other so on. As I am repeteadly saying the only way in this present condition (ie. Kaliyuga) is to go to a hospital. One may use mantra methods while he is on the way to hospital, or while hospital is not avaliable or antidote is finished, or when hospital can't treat due to seriousness. I'm not doubting the potency of mantras by emphasizing on the word Hospital but I'm doubting the potency of person who chants mantra.