Making sure you have a Pokemon that can use Flash is recommended before proceeding too far, because the basement is very dark and requires Flash to see very far. Surf is also required to reach the basement.
Anyway, there's a Hiker that can be fought near the entrance, as long as you climb up the steps up to the nearby ledge. He uses a Hippowdon L61, a Marowak L61, and a Rhydon L61. Just past him is a Max Repel.
Surf over to the west to pick up an Escape Rope and fight a Battle Girl, who has a Medicham L62, a Hitmonlee L62, and a Mienshao L62. Follow the path to the south and then to the east until you find stairs leading down into the basement.
Wellspring Cave B1F
The basement is very dark, so now's the time to use Flash to light it up. Head down the steps to the east and then head north, going east until you run into the Black Belt. He's got a Sawk L62, a Conkeldurr L62, and a Breloom L62. Surf across the water by the Black Belt to reach a Max Revive, but don't jump down the ledge near it unless you want a trip back to the beginning!
North of the Black Belt is a Battle Girl. She uses a Primeape L62, a Hitmonchan L62, and a Scrafty L62. Just above her are some steps leading up, although the path branches off in two directions. Take the past west from the steps and follow it south to reach TM47 (Low Sweep), which is a Fighting-type attack that lowers the foe's Speed afterwards.
Go back to the branch point except go east instead. Take the path south in order to find a hidden Carbos. Head back except head down the stairs just past that southern path to pick up the Protector, which evolves your Rhydon into Rhyperior when you trade it while it is equipped. Nifty.
Head north of the Protector until you find some steps leading down to a Black Belt. He's got a Throh L62, a Hitmontop L62, and a Machamp L62. Surf up to the north and you'll find a Veteran. He has a bunch of Pokemon on his team! He's got a Gyarados L62, a Druddigon L62, an Excadrill L62, a Gigalith L62, a Crobat L62, and a Snorlax L62. After taking down his legion of Pokemon, head north and pick up your prize: TM52 (Focus Blast), a powerful — although inaccurate — Special-based Fighting-type attack with a respectable 120 power. If only it didn't have 70% accuracy, it'd be nice, but since several non-Fighting-type Pokemon can learn it, including many Psychic-types, it is still handy in certain situations.
East of the Veteran is a hidden Max Revive, too, which you should definitely pick up. Additionally, there's a hidden Ultra Ball on the patch of land to the west of him, although you'll need to Surf over to it if you want it.
Well, that's all there is to do in the Wellspring Cave. To get out quickly, you could always use that Escape Rope, haha. Or if you feel like taking the scenic route, head south from the hidden Ultra Ball along the water and you'll reach the entrance to the basement.