How do I get to the second gym in Pokémon Alpha Sapphire?

At this point in the game, you still haven't received the HM for Surf, so you can't reach Dewford Town yourself quite yet.

If you follow the story properly, you'll eventually get to board a ship with which you'll reach not only Dewford Town (where the second gym is), but also Slateport City later on.

Basically, you need to recover a package which was stolen from Devon Corporation, after which you'll get to speak with Mr. Stone.

"I'm Mr. Stone, the president of the Devon Corporation. I heard just now how we have quite a bit to thank you for. So you helped out my staff not just once, but twice! There's a favor I'd like to ask of a reliable Trainer like you. [...]"

"I want to ask you to deliver this Letter to a man named Steven. You should find him in Dewford."

"You'll have to cross the sea to reach Dewford, of course. I'll ask Mr. Briney, the old sailor who lives in that hut on Route 104, to help with the crossing. Go find him, and he should agree to help you. I'm counting on you, young man/lady! If you're ever unsure of what to do, just come and talk with me again!"

Only after you deliver the letter will Mr. Briney let you sail to Slateport City, from where you can reach the next gym.