I'm ak521! I've been playing League of Legends since Season 6 and I believe that TFT is an awesome and fun gamemode. If you haven't given it a try, I strongly encourage you to do so!
This guide will teach you the following:

• Basics
• Popular Strategies (e.g. hyper-rolling)
• 
  
  This will get more colorful soon!

BASIC GAMEPLAY
First, there is a champion "carousel"- each of which has a single item on them. I will discuss the best picks later in this guide under [WIP]. If you do not pick one, you will get a random one from one that is left.
For the first three rounds, you are against monsters (AI). You are guaranteed to win Stage 1-1 (first game) with any champion despite the item. Your next two rounds are also guaranteed wins if you place the correct number of champions in (2 for 1-2, 3 for 1-3). Defeating these monsters will give you items, gold, or champions (there are mystery "?" orbs which you gain bonus movement speed towards- you must walk over them to gain the bonuses.
You can buy champions in the shop at the bottom with the currency, Gold. The champions have at least 1 origin and at least 1 class, and you can place them on the board to activate certain synergies. Champions fight on their own, normally prioritizing the closest champion with a few exceptions (assassins jump to the back before doing this). You can also spend your gold in re-rolling (will discuss later under [WIP]) or to buy XP.
You gain 2 XP automatically at the start of each round. Gaining a level allows you to place an additional champion on the board.
After the AI rounds, you will start to battle other players. Winning will not give you anything, but losing will cause you to take damage (out of 100 total). The damage is increased base on the amount of champions the opponent has left, their level, the champions' star-level, and the champions' cost.

**WIP GUIDE, MORE TO ADD SOON!**
Vayne Nasus Jax Aatrox Kindred Soraka Master Yi
When to make This is an extremely strong team composition. You want to (A.) have a spatula by the end of the Krugs (B.) have at least 2 two star light champions by the end of Krugs and (C.) have a viable early game synergy with champions to switch off with.
Synergies

• Light x6
• Mystic x2
• Shadow x2
• Blademaster x2
• Ranger x2

Mandatory Items

• Vayne: Giant Slayer Guinsoo's Rageblade
• Master Yi: Talisman of Light

Early Game - save up Gold. If you do not have a spatula, it may be worth taking hits just to get it.

Mid Game - Find a viable synergy. Lights often are not very strong until you get 6 lights. Try Shadows, Blademasters, or Rangers. Glacials work, as well.

Late Game - Be sure you know when to make Level 7. You should have at LEAST 1 level 3, preferably 2. You can use Yasuo] while you are waiting for Master Yi If you get a light Lux, try to put her and Yorick in ASAP for an almost free first place.

Kog'Maw Warwick Volibear Dr. Mundo Skarner Olaf Twitch Ashe
Synergies

• Predator x3
• Poison x3
• Berserker x3
• Glacial x2
• Ranger x2
• Crystal x2

Mandatory Items

• Kog'Maw: Guinsoo's Rageblade Spear of Shojin Luden's Tempest
• Twitch: Giant Slayer OR Sword Breaker (at least one on-hit item)

Early Game - Try to make use of the Predator bonus. You can use Ornn Volibear and Warwick to get a viable composition temporarily, or possibly a Braum

Mid Game - Try to force in Predators and Berserkers. Also try to make Kog'Maw 3*. You can replace Volibear if you get a Black Cleaver and put it on Warwick.

Late Game - Roll to level up your composition and find the last few champions you need.

Zyra Malzahar Vladimir Kindred Veigar Annie Master Yi Nami
Synergies

• Shadow x4
• Summoners x3
• Mystic x2
• Mage x3
• Inferno x3
• Ocean x2

Mandatory Items

• Malzahar: Mage's Cap, Rabadon's Deathcap, Seraph's Embrace
• Brand: Jeweled Gauntlet

Early Game - Find Summoners and Mages. I would recommend saving the Mages in the back till you have enough space for them. Try to make Mage's Cap. You can substitute Vladimir with Syndra.

Mid Game - Play 4 Shadows and 3 Summoners. Substitute Sion until you find Master Yi. Try to make Mage's Cap. If you're able to, use 3 mages instead of 3 summoners.

Late Game - Put in all of your synergies. If you reach level 9, you can also put in another blademaster to get that buff, too. Alternatively you can put Blade of the Ruined King on Kindred