Dungeons are relevant if you do Mythic+.
Mythic+ is basically just a scaling dungeon difficulty.
If you complete any Mythic dungeon (including 0, which does not take a keystone), you’ll get a keystone if you don’t already have one. Inserting the keystone into the pedestal at the start of the dungeon activates the difficulty labeled on the keystone, as well as a timer. If you finish the dungeon before the timer runs out, your keystone is upgraded to a higher difficulty (based on how much time was left). Rinse and repeat. The soft cap for loot is currently Mythic+10, but will be bumping up to +15 in the next patch IIRC.
You cannot use the dungeon finder tool for this. You have to manually form or join groups. You can use the Premade Group Finder tool to do this, but players have the say-so on inviting you, so it may be frustrating if you’re behind on gear and experience.
Dungeons below Mythic+ are mostly irrelevant right now. They’re sometimes useful depending on the time period/expansion, but right now they’re not.
Raiding isn’t super relevant right now unless you want to push Mythic difficulty, which is by far the most demanding standard content in the game. Most people won’t bother.
You’re better off focusing on World Quests (mostly for Emissary rewards – open your map, there should be some circles on the bottom left indicating these) and Benthic Gear (obtained from Nazjatar via Mana Pearls, which are earned from some quests there as well as commanders that get marked on the map and world quests specific to that zone).
Benthic Gear can be upgraded to ilvl 425 via Mana Pearls. Heroic Raiding is only ilvl 430, which is why I say Raiding isn’t super relevant right now, since Heroic is the last difficulty with decent participation numbers.
The importance of Benthic Gear vs Mythic+ Dungeons and Raiding will likely decrease in the patch coming in a bout a month, as new content comes with new ilvl, and AFAIK Benthic isn’t changing and there isn’t a replacement system for it. There will be catchup gear that can get you to ilvl 400 quickly, but Benthic goes well beyond that already.
Benthic is also the current catch up, mostly because buying the ilvl 385 pieces are very cheap. 5 pearls. It’s upgrading that’s expensive. If you boosted to 120, the base 385 pieces are irrelevant, but you’ll still need them to upgrade them to something that IS.
If you don’t know how to get to Nazjatar yet, check the big Alliance boat in the Boralus harbor, assuming you’re Alliance. Orange (not yellow) quest markers are what you should be looking for. I find it unlikely that the character boost doesn’t let you start right away with Nazjatar, so if you boosted you probably just missed the quest.
If that’s wrong, no big deal, just start questing in each Kul Tiras zone until you get friendly rep. If you boosted you skipped that. There’s a scouting map thing in the Harbormaster’s Office that should let you choose zone questlines. Or potentially a tour quest from an NPC on the docks in the market district. I’m not entirely sure where you need to pick up from on a boosted character in regards to standard questlines. It’s probably one of those two things.