As you arrive in the city and approach the Poké Mart, May will emerge from the Mart, marvelling at the size of the city. She’ll also note to you the Pokémon Gym before leaving. Anyhow, head for the Mart first; inside, speak with the man next to the counter to earn TM54 (False Swipe)!
False Swipe is an invaluable move when attempting to capture Pokémon - well, non-Ghost Pokémon - as it cannot KO a Pokémon, lowering its HP as low as 1, which greatly increases the odds to capture it, especially when in conjunction with the Paralysis or Sleep ailments!
From the northern Pokémon Center, go west one house and speak with the man within. He will give you HM01 (Cut). HMs are a bit different from TMs; while they do teach moves, so long as the proper Badge is obtained, the HM move is usable in the field!
Cut is a pretty bad Normal-type move (50 Power) but it does let you cut down trees once you get the Stone Badge, which is obviously helpful.
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In the building east of the Poké Mart, you can find the Pokémon Trainers’ School. Feel free to brush up on some basics there; you can also speak with the man at the desk to get the Quick Claw .
The Quick Claw is well, it’s an okay item for Pokémon who are only about average, but Pokémon that are above average won’t really need to bother with it. Plus, it only works 3/16 of the time (so long as the opponent isn’t using a a higher priority move like Quick Attack) which isn’t too reliable.
Next enter the house north of the Center and the Mart, the building which houses many of the Devon Corp’s employees. Go to the second floor and speak with the Ace Trainer there to get a Float Stone , an item that halves the holder’s weight. (It’s actually not too useful; very few moves consider weight.)
Next, head northeast to the Pokémon Gym, and enter the house to the east of it. If you speak with the little kid there, he’ll complain that people laugh at him whenever he says his Makuhita’s nickname, and wants you to give him a Slakoth in exchange to avoid the shame.
It’s completely a trade upwards for you, so you might as well. Makuhita can be a decent Fighting Pokémon, though there are quite a few better ones and Makuhita can be naturally caught later on.
Anyhow. Back to business. Go west from there and north along the road for a bit and head onto Route 115 to the north if you want to, or Route 116 to the east.
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Take a look at least at the latter Route to build levels for the upcoming Rustboro Gym, particularly if you’re using Torchic. If you’re using Torchic and don’t have Wingull or Shroomish, you’ll want Torchic to hit Level 16 soon to evolve into the Fighting/Fire Combusken.