So I started to play this game and I want to play it as a Priest, party because in my opinion a game named Inquisitor should be played as one and partly because I like the mage class.

So now I have the first quest, killing bats, but my priest can't handle them. Magic doesn't really hurt them, bashing them works a littly better but he is constantly out of stamina and he might bashes three of them and than gets killed pretty quickly. I expected priests are pussies but how can you succesfully play a mage character if the spells don't work?

Obviously I'm doing it wrong.

So what should I do?

Equip the three seals you start with in your quick slots and spam those at the bats, fire is AOE so thats pretty useful.
Otherwise you need to level up your wisdom skill or whatever class of magic you want to use to make them not fail as often (im not sure which one stops spell failure, clarification anyone?)

The Priest is weak and has limited options in the beginning and early middle parts of the game due to the tendency to fail often in spellcasting, but will progress to become a very powerful character with the ability to clear out an entire screen of enemies within seconds towards the end of the game.

In the beginning against bats, bashing them is ok as you will probably only have weapons from stashes and dead bodies scattered at the edge of Hillbrandt. Try not to attract too many bats on you and fight the bats individually or in as small a number as possible. Use the quicksave and quickload, save and restore game functions often as they are very useful for scouting groups of enemies and after a battle where you need to wait for your stamina to recharge. If there is a large group aggroing you, use the seals as suggested as they do more damage than any weapon.

Health, stamina or mana drain and cause negative condition(disorient, poison, paralysis, slow etc.) is very common in almost every hit on you or companion made by many enemies in this game so playing conservatively and waiting for stamina to recharge after every battle is a good strategy as running out of stamina will reduce your rate of attack and prevent spellcasting.

Yes, skill level and rank in Wisdom and the class of magic which the spell belongs to will help to reduce the chance of failure in spellcasting. Later on, your alignment and a certain spell at Magister level can also help to reduce the chance of failure in spellcasting.