There are lots of trainers on Route 3. This is probably the best time to train your Pokemon. They will gain lots of XP and some levels.
Finally, it'll all be over. Just continue up the Route to get to the entrance of Mt.Moon.
NOTE: There is a PokeCenter if you need to use it from all those trainer battles.
I'd also get repels since the wild Pokemons in the caves are VERY annoying.
Walk to the right to advance through the cave.
Turn right and past this ladder.
Head straight up then left.
Now, down and to the left again.
And finally up again.
There'll be a Moon Stone at the back so be sure to pick it up.
And to the right will be a ladder that you go down.
Just follow the tunnel to the ladder.
Now, head downstairs and talk to the Rocket Team member and battle him.
Head up the stairs behind the Rocket Team member and follow the trail.
When you get to the end of the traile, head straight.
You will then find a super nerd analysing the fossils he found. After defeating him in a battle,he'll give you a fossil. Pick the one you want. NOTE: Depending on which you one you get, you'll get a fossil Pokemon when you get to Cinnabar Island and have it turned alive. Dome Fossils contain a Kabuto and the Helix Fossil regenrates a Omanyte.
After you've done that, head to the left where will be a ladder.
Here, you'll meet Jesse and James, members of Team Rocket. You'll have to defeat them in a battle.
After going down the ladder after you defeat Jesse and James, go to this ladder to exit Mt.Moon/ Now, off to Cerulean City!