Note: This page of the walkthrough is for the first time you enter Wellspring Cave. If you want a guide for the entire cave after you have Surf, go to the Wellspring Cave Revisited page of the walkthrough.
It is a good time to save right as you enter. When you're ready, walk toward the two Plasma Grunts which will lead into a battle.
Two more Plasma Grunts will come to battle, but this time it will be a double battle with you and Cheren on a team. Cheren will use his starter. This battle shouldn't be hard, but it can be difficult if you are unlucky. You have no control over Cheren's Pokemon, so if one of the Plasma Grunts' Patrats use Bide and Cheren decides to attack it, you could be in trouble. With any luck you will attack a Patrat and then Cheren's Pokemon will finish it off.
After the battle there will be some talking. The Plasma Grunts will leave and Cheren will go back to Route 3. You can walk west and make your way around the rock to the item ball which contains TM 46, Thief. That's all there is to do here now, so walk back outside to Route 3.