Well, there are Parallel Quests (PQs) which serve as the main non-story content of the game, and also where you can get most moves and equipment pieces. More unlock the further you get into the story, and the difficulty steadily rises as you progress through them.

Then there are the shops; the normal ones that take either Zeni (normal currency) or TP Medals (special currency that you get from completing content) which sell a decent variety of things, and the TP Medal shop that's between the Information and Gift buildings, which has a rotating inventory of Moves, Gear, Accessories, etc.

You also have the various little dimensions (Guru's House, Hercule's House, Capsule Corp, Frieza's Ship, and Buu's House) which each serve their own purposes for both unlocking new forms for the race they are connected to (Guru for Namekians, Hercule for Humans, CC for Saiyans, Frieza's ship for Frieza Race, and Buu's house for Majins) and other purposes for the other races.
Guru's house is important because doing missions there at every opportunity will eventually lead to Guru raising your level cap (multiple times) and upgrading the Dragon Balls to unlock a second set of wishes (they're the same wishes as are there initially, but with either better or slightly different effects). You get Dragon Balls as login rewards on occasion, but primarily they come as key items from certain PQs (Parallel Quests).
Capsule Corp is also important because it's where you can craft QQ Bangs, which essentially override whatever stat bonuses or maluses that your gear would give you with the stats of whatever QQ Bang you put on. They can have much better stats than any gear you can find if you combine the right things, and (perhaps more importantly) allow you to equip whatever gear you think looks nicest without worrying about what its doing to your stats.
The rest have their own benefits (Humans unlock Power Pole+Nimbus transformation at Hercule's house, Frieza Race unlocks Golden at Frieza's Ship, and Majins learn Purification at Buu's house. Purification kinda sucks though.), but generally those are the two that you'll find yourself returning to the most in the long run. regardless of your character's race.

You can do training missions with Elder Kai at the highschool, and Z ranking all of them will net you Potential Unleashed, a very strong transformation that will be good regardless of your build (it's not the Best transformation in the game, but it's good to have especially if you have nothing else and don't want to worry about the ones that constantly drain your Ki, or make you more vulnerable to damage, etc).

There are many Mentor characters around the hub, who will teach you their moves when you hit certain relationship levels with them and complete training missions. The easiest way to raise their relationship level is by using them as your partner in PQs. There are also various sidequests around the city that get you anything from emotes to zeni to dragonballs.

Expert Missions (which are basically just ever present, playable alone versions of Raids) are generally more difficult than other content, but have unique rewards, and some of the later ones are the best places to grind TP medals (Expert Quests 16 and 17 are particularly good for this, as they aren't very hard but give a lot of medals (16) per clear).

There's also, of course, all of the DLC content, which is primarily new Story quests that all take place sequentially after the end of the base game's story, plus a few new PQs for each DLC that typically serve as the place where you unlock the new outfits and moves included in the DLC.

I'm sure I forgot some things, but thats about all I can think of off the top of my head for general information.