I came to Destiny 2 never having played the original, and had a lot of the same questions starting out. But essentially, the best advice I can give you is don’t think too hard about things and just roll with it. The shooting/combat is fun, and there are no penalties for failure, other than some wasted time, which if the shooting/combat is fun, then it really wasn’t wasted. The game has many, many huge grinds if you let it. First there’s xp to level up, and then upping your power, and oh yeah, two more classes to repeat all this on. It’s kind of nuts, so only engage with it if you want to, don’t feel you have to.
As to specifics: There are three elements that influence weapons and armor, and importantly shields. If your target is surrounded by a bubble of orange, for example, then you want to hit it with a solar weapon if you can, as bringing down the shield will explode it and potentially deal damage to surrounding enemies. In the early game, any weapon will work, but solar will work a little better. Endgame activities, solar might be the only effective way to bring down that shield, but it’ll be a while before you reach that point. For armor, the element type will determine some of the mods it can slot. Be aware of it, but don’t worry about it too much, it only matters much later.
Some activities have matchmaking, which is location based, so it’s trying to put you in lobbies with folks geographically near to you for a better connection. If you’re playing at 3am on a Wednesday, this might be a problem, but if it is, it will eventually give up, launch the activity, and then if it finds someone later it might add them. Not sure how many new players the game pulls in regularly, and older players don’t have a reason to revisit early content, so I don’t really have any advice here.
Don’t worry about upgrading things in the early game, mostly focus on weapons you find that you enjoy using or that fit your play style. A lot of the fiddly bits are hidden, and you can most certainly look them up online, but early on just focus on the fun. Learning the difference between a gun with kill clip and another with rampage and how they’re different and why you might want one over the other can wait. It’s helpful, but it shouldn’t be a priority.
The Cosmodrome is a location on Earth, and yes, other planets are involved eventually, with much better and more interesting artwork/geometry.