I'm a SF fan and I heard a lot about Traveller, but I still don't know how the game plays.
The game system is incredibly smooth. As mentioned it's mostly roll 2d6+mods vs a target number, usually 8. Damage is tracked as ability score loss. More often than not it's played as your typical space-truckers style game. There are options for exploration- or military-focused games, but those are less common. Not quite the criminals of Firefly or the bounty hunters of Cowboy Beebop, but those are really close.
In the space truckers variant it's mostly: need money and fuel, find a job, work the job, deal with complications related to the job, complete the job, get paid, fuel up the PCs' stomachs and the gas tank, and back to one. Different referees focus on or emphasize the spaceships & spreadsheets aspect to different degrees. Some are hyper focused on the math, others handwave most of the money-based paperwork.
What kind of characters you can create?
Traveller was the first game to use lifepath character creation. So it's a mini game unto itself. In the old days you could die in character creation. It's still there but as an optional rule. There are no classes or levels, Traveller is a skill-based game. Skill packages come with careers. The current edition is Mongoose 2E, which has 12 careers in the main book, each with three subcategories. The core book has agent, army, citizen, drifter, entertainer, marine, merchant, navy, noble, rogue, scholar, and scout. Other books have additional careers, including criminal and psychic. Based on your lifepath you get benefits, cash, ship shares, and debt.
What kind of adventures and stories should I expect?
Your typical space trucker and space opera style adventures. You jump to a new system, encounter a derelict ship, explore it hoping for salvage, only to discover there's something alive over there. You jump to a new system, encounter a planet of backwater folks barely scraping by and they need your help. Think Star Wars without the Force and far fewer aliens. If you played any WEG Star Wars back in the day, Traveller is a tramp freighter campaign with slightly less wild technology.
What do I spend most of my time doing in a game session?
Depends on what the referee and players focus on. Roleplaying, combat, exploration, dealing with complications, making plans, watching plans go up in smoke at first contact with the enemy, etc.
If you think Traveller is a good game, what makes it good, and what should I be looking for in a game to enjoy it?
I do think it's a good game. It's a slightly more grounded space opera game. I really, really like that. You don't press a button and jump from one side of the galaxy to another. You enter jump space and travel for two weeks and you're either 1-, 2-, or 3-parsecs away. It focuses on the minutiae which really helps with immersion, for me.
If you want an incredibly detailed deep dive, check out Seth Skorkowsky's channel. He has two relevant playlists. One is an
overview/review of the game and system. The other is a
series of adventure reviews. If you want more info on what a typical adventure looks like, pick an adventure review and watch that.