I just started my free trial two days ago. I've done two quests so far, but I'm still so overwhelmed by all the stuff on screen and I don't know how to talk to other players.
Honestly, my best recommendation is to try and tune out sidequests, and focus on moving through MSQ - particularly if you're feeling overwhelmed by the amount of Things to Do.
All that other stuff will be there for when you're ready to branch out.
Better equipment you may be able to buy from your local City-State vendor.
How do I find people to play with? Am I allowed to play with other people? I tried the invite to party feature but it said not available in free trial.
Here's a comprehensive list of the things you can't do on the free trial - unfortunately, yes, that includes setting up parties. However, other people can invite you to their parties, and you can join party finders and you can use the duty finder. You just can't be the lead.
But the core gameplay will have you thrown into dungeons with other players, whether you've partied up prior to queuing for the content or not.
The one on top is the main story quest. This will yield you most of your exp to level up and progresses the story. Most content in this game is locked behind story progression. It's also kind of a long ass tutorial.
The second one ("Trial by Wind" in the picture) is your job quest. This will grant gear and sometimes unlock abilities.
A third icon to look out for is a blue/purpleish quest icon with a little plus next to it. This is a quest that will unlock some kind of content or feature for you, but you can always just come back to those later when you're ready.
The duty finder will hook you up with other random players to run content that requires a party, so you don't necessarily need to make friends or preform parties in this game to play 90% of content. But if you'd like, as a free trial user you can still join linkshells (a chat group). If you do want to form a party, but the other player(s) is also a free trial user, its common for trial players to ask in shout chat (/shout talks to all players on the map) for a subbed player to invite them to a party and then leave.