You'll enter Striaton City by a nice garden, with water ponds and shrubs all around. Surf across the water to the left and you'll reach a Fisherman who will give you a Big Pearl for talking to him. South of him is a Max Elixir. In the bottom-left area, there's a hidden Full Restore, while there's a Yellow Shard in the bottom-right area. Lastly, in the northeast part of the garden, Surf across the water and then land over by the gate to find a hidden PP Up.
The first building you'll come up across is the Trainers' School. There's a School Kid inside on the left-hand side of the room that will quiz you on which berries cure which conditions. You'll want to answer the top answer for the first question, then the bottom answer. He'll give you a Persim Berry for your trouble, which cures confusion when held. The Ace Trainer inside asks you questions about battling and you can answer, although being in Japanese, it's probably won't make much sense to you, either.
The little girl inside of the house south of the Trainers' School will give you a Funfest Mission when you talk to her, though you may have to select the middle option.
Inside of the Pokemon Center, you'll be able to buy two nifty TMs from the Poke Mart: TM04 (Calm Mind) and TM08 (Bulk Up), each for $80,000.
If you have the rare event Pokemon, Shaymin, in your party but are in need of the Gracidea flower used to change its form, look no further: talk to the lady near the desk of the Pokemon Center with a Shaymin in your party and she'll give you the Gracidea, which can be used on Shaymin to transform it into its Sky Forme. It can only be used during the daytime, and only works if Shaymin isn't Frozen.
Keep in mind that Shaymin was only available in the previous generation of Pokemon games, and only as a special Mystery Gift event, so you will need to trade for it if you missed out on the event — and they tend to be rather rare.
You can find Clyde stationed right outside of the old Pokemon Gym, although it has been converted entirely into a restaurant. Get to that in a moment, because there are a few more things that can be investigated in town.
The house directly south of the restaurant is Fennel and Amanita's old house. They've moved off to Castelia City, so it's just the family that remains — well, that and an Up-Grade upstairs.
Right beneath that house is another one. The man inside will give you a Great Ball when you talk to him.
To the south is Route 2, which you can check out essentially whenever you want. Hug the fence to the west, though, so you can find a Timer Ball sitting around. Behind the two buildings you just checked out, there's a Roughneck who will give you a Dusk Ball after talking to him. Then, to the northeast is the entrance to the Dreamyard, which is a great place to check out and is even where you can find your next legendary Pokemon!
However, before you run off, you should heal up your Pokemon and check out the restaurant. Inside, you'll find the retired Gym Leaders of the Striaton Gym — Cilan, Cress, and Chili. Talk to one of them and they'll ask if you'd like to join them in a Double Battle against the other two, so decide which of the three you would like to team up with, and then get ready for a fun battle!
||PKMN Trainer Cilan||$16080|
||PKMN Trainer Chili|
||PKMN Trainer Cress|
Obviously, you will be fighting only two of these three, while the odd one out is the one you chose to be your partner. Each of the elemental monkeys has a Life Orb equipped, meaning they'll do a sizable amount of damage. You can fight the trio once per day, regardless of who you choose as your partner.
Additionally, you can talk to the lady up on the stage in the back of the restaurant to play a game. She'll introduce three Pansear to you and then one of them will jump up and down. She'll tell you that it has a Big Mushroom and it's yours if you can keep track of it. You have to keep an eye on which Pansear it was that is holding the mushroom — in other words, it's just like the old shell game. Watch it carefully and select it to be rewarded with a Big Mushroom, courtesy of a Panpour. You can do this once per day.
Well, that's all there is to do in Striaton City. From here, you can continue southbound towards Accumula Town by following Route 2, or you can check out the optional Dreamyard to the northeast, which contains a legendary Pokemon for you to catch.