You need to earn 4 badges in your starting region to unlock interregional travel.
After that you can go to the port city of your region (Vermilion/Slateport/Castelia/Pastoria) and board a ship.
If you don't like it there, you can grab the regional starter and then whine at your mom — she'll yeet you back to your starting region — if you don't have 2 badges in the new region yet (4 for Sinnoh). If you do, just use the ship again.

All items except HMs and key items go with you.
When you're visiting a new region for the first time, your mons are boxed and you get a starter. You can take them out and use them as soon as you reach a PC — if you have enough badges to use them in the new region.
https://pastelink.net/levelcaps — check this link if you're not sure.
After that your mons just stay in your party.

And yes, you can, for example, bring mons from Unova to Kanto — this is a single game world.