You don't need to go into that cave yet, as you will not get to use Strength until the...6th Gym I think it was? Edit: Oh, and the Strength HM is at the top floor of the Lost Tower, but you haven't reached there yet. So don't worry.
As far as the 3rd Gym goes, your next destination is Hearthome. Go south from Eterna (where the 2nd Gym is), down Cycling Road, then east into Mt. Coronet. Crossing through there will lead you to Hearthome. Just follow the events from there and you'll make it to your next Gym battle in Veilstone.