Adventure is a nice ship. Sailed her twice last year out of Galveston. Doesn't have all the bells and whistles of the newest ones but still a good introduction to the line. On deck 5 is the Royal Promenade. Sort of the main thoroughfare of the ship. Like a mall that runs down the center of the ship. Here you'll find shops, an English Pub, Cafe Promenade which is where you can grab complimentary sandwiches or pizza.
Don't stress over dining room dress codes. On a 5 night cruise you'll only have 1 formal night. If you don't feel like packing a suit and dealing with that then just go with sort of a business casual look - collared shirt and pants (even jeans as long as they're not all ripped will be ok). You'll probably see shorts in the dining room but usually the only thing that will get you turned away is swim attire, flip flops, etc.
There is an ice skating rink on the ship. It's on deck 3 I believe called Studio B. This is where they have the ice show but it will also be open several times throughout the cruise for open skate. If this is something you'd like to do, bring long pants/jeans as that's part of the requirement.
Another odd thing about going out of Galveston - they'll have a very limited drink menu available until they reach international waters ~12 miles out. So from boarding until around dinner time you'll only have a few drinks and mostly domestic beers (and Shiner) available.
If you don't have any excursions planned then both stops are good typical cruise ports with lots of shops and places to eat and drink. Costa Maya is a nicely laid out port with a very large pool and bar in the center of the port.
You said it's your first Royal cruise, but first cruise overall? You'll want to download the Royal app. When you leave the dock, put your phone on airplane mode and turn on wifi - you'll be able to use the app to see the schedule of events around the ship, make reservations for dining, etc. If you want to do anything else online you'll need to purchase the internet package.
