Just fighting all the trainers and continue on up. You can really wreck shop if you have a built-up Fearow or similar flying Pokémon, who also has a Dark attack (like Pursuit). The Pokémon Tower is really linear and there aren't really any tricks to getting through. It's devastating against the Ghost enemies. Since you've got Dig, be sure to teach it to one of the Pokémon in your group. All Ghosts are also dual Poison types, so Ground, Psychic and Bug attacks work especially well. Instead, use whatever Psychic attacks you've got.
Since this is your first encounter with ghosts, here's an important tip: don't use Normal or Fighting type attacks. Turns out those hostile apparitions are just Haunters and Gastlies. The Silph Scope will turn the unknown specters you would otherwise encounter into, well.
If you've found the Silph Scope in Team Rocket's Hideout, you can now safely enter the Pokémon Tower in Lavender Town.
