Your next destination is Union Cave. You can exit the Ruins of Alph from the east, or exit Violet City from the south, to get to Route 32, which leads to Union Cave.
Make your way south, catching pokémon if you wish, and battling trainers. Be sure to look around for items; there are some good ones here.
After passing by some water and a raised train line, you will find a Pokémon center. If it's Friday, you will find Frieda north of the Pokémon Center. She will give you a Poison Barb.
Go into the Pokémon Center and talk to the people there; one will give you the Old Rod. You can use it to fish in any body of water. However, the Old Rod mostly catches Magikarp, which are really weak pokémon that everyone makes fun of. But the Old Rod does sometimes catch better fish, so give it a try.
At one point around here you will find a guy trying to sell a SlowpokeTail for 100 dollars, but he won't actually sell you one.
If you use Rock Smash on the rock south of the Pokécenter, you get a Shell Bell.
Now you can go into Union Cave.