Can someone give me a brief rundown of what I should be trying to do while playing TFT?

Grab a 2-cost champion from the first carousel. Preferably one with Recurve Bow or Needlessly Large Rod. Fight the first round. Sell the 2-cost character and buy 4 tier 1 characters - buy any champs that also have multiples (like if you see 2 Nidalee etc). Rageblade and Shojin are the best items in the game. Build them on champions that make sense. Don't build Shojin on Graves, for example (Graves doesn't use mana making the Tear of the Goddess and the Shojin passive useless).

Try your best to find matching factions. Synergy is key.

Put items on your carry and build the right items that you have access to. Ex. If your opponents are building assassins, and you can build a Phantom Dancer, then build it.

Prioritize items in the carousels unless you have a lot to gain (like a tier-3 champion) from taking a champion over an item.

Don't waste tons of gold refreshing for more champions. Save your gold if you can (you gain 1 interest up to 5 for each 10 gold that you have - EX 1 gold if you have 10 or more in the bank. 5 gold if you have 50 or more etc).

Position your champions reasonably well. Fights have a lot of RNG but you can manipulate this somewhat with proper positioning. Tanks in the front, carries in the back, and that kind of thing.

That's basically the foundational stuff. Become familiar with items. Don't underestimate spatula items. They can combine to give you that last member of a faction (you can Golden Spatula + Giant's Belt to get that last glacial character you need for the max bonus, etc).

Pick ashe, vayne, varus, and kindred as they appear

Get 2 star mordakiser and braum

Replace braum with senjuani when she becomes 2 star

Put rapidfire cannon, rage blade, and phantom dancer on as many of the rangers as you can, and you just win 90% of all games

Carrosell you pick only recurve bows or needless large characters unless there is a full variant of one or if you see an early kindred with like a chain vest thats okay too

Elise can carry you the first 2 minion waves so if she appears on the first pick you can snag her to eco the first 2 (losing till kruggs is fine so long as you beat the krugs when you get there)

Pray to our lord and savior RNGesus

https://i.kym-cdn.com/entries/icons/original/000/018/026/nUQACjz.jpg

Ill give you the answer I give all my friends when they ask for general advice on this game: it depends

Just play a lot, figure out what works, what doesn't, look up guides, watch streams. There's WAY too many things you should be doing to list here. Even general advice is situational.