In Union Cave, there are trainers and wild pokémon. If you head to the northwest, you will find some stairs leading down. You can't do much here, but there is a TM39 Rock Tomb and an X Defend. This is a dead end, though (for now) so go back upstairs and head south.
Eventually, after going south, you will reach the exit of the cave, as well as an Awakening item. Exit the cave.
It's raining here, which will weaken fire-type moves and strengthen water-type moves. It will also inspire some funny reactions from different pokémon when they walk with you, so try that out. Get the two apricorns, fight in the grass if you wish, and battle the hiker. He'll give you his number.
Head west to get to Azalea Town. A team rocket dude is being all mean to someone who runs away. You'll find some tail-less slowpokes in town, too. Buy stuff at the mart if you wish, then go to the house in the northwest and talk to Kurt. He'll be really upset that team rocket is cutting off slowpoke tails, and he'll rush over to the slowpoke well to try to get rid of the bad guys.
If you wish, you can catch some new pokémon in the forest west of town, but you'll reach a dead end there very soon, so go back into town.
Go to the slowpoke well to find that Kurt has gotten rid of the guard who was blocking the way. Now you can go down into the well.