When you get to Violet City, your best friend will give you a Vs. Recorder, which allows you to record battles with online trainers or with Battle Frontier trainers.
In town, you can trade a bellsprout for an Onix.
You can't get to the gym leader until you have gone through Sprout Tower, so go to the tower to the north. You will encounter Sages here who will have battles with you. You can also find several useful items, so look around. The Sages for the most part use Bellsprouts, so don't send any water pokémon or rock pokémon out against them.
At the top of the tower, you find Silver leaving the tower after receiving a TM, which the Elder awarded Silver for defeating him. Now, it's your turn to fight the Elder, who has two Bellsprouts and a Hoothoot.
For winning, you get TM70 Flash. You can teach that to a pokémon, although not all pokémon can learn it. When you use a TM, it will show you which pokémon can or cannot learn it. If you teach Flash to a pokémon, it will be able to light up dark places, like Dark Cave for example (but even if you go there and use Flash right now, you won't be able to do much.) In battle, Flash will lower the accuracy of your opponent.
You can get an Escape Rope behind the Elder, which you can use to get out of the tower.