Thanks!
Opening Basics for Teams

Very good description. Just 2 notes:
1. You do not show how blue and green can mate yellow or red on the first moves. If red (or yellow) does not put their bishop to h2 (g13) and knight to f3 (i12).
2. This phrase is incorrect: "Blue and green must follow suit and move the queens in front of their kings. This is CRITICAL for blue and green." There are other possible moves. For example, bishops can go to b7 (m8) instead of queens. Most of players now move their queens to those white squares, but it is a matter of fashion, it allows to make 3 to 5 first moves automatically and start to think later.
A bishop at b7 (m8) may be more dangerous in some cases. It can be sacrificed with a check to win red or yellow queen. But if you blue or green move their bishop there, then they are going to spend more moves to activate their queen.
Thanks for your input! I agree that the opening is really just like you said, a matter of fashion. Moving the bishop to b7 for example is another variation which I did not show (I must admit I'm biased against that move as it goes against my main point of getting queens out early into the game and giving them room to be activated quickly haha). And yes I neglected to show some of the ways red/yellow might get mated. I didn't want to discuss mating patterns much in this post as I could serve up and entire discussion on that in another post for later. I instead wanted to focus more on some of the opening moves while giving just a couple of more obvious examples where things could go wrong if some moves are not played accurately.
4 player Teams is a totally different concept than that of 4 player FFA. It involves lots of tactics which changes the way pieces are moved on the board. Here are some useful opening tips for beginners who might not understand the difference between FFA and Teams.
In 4 player chess, red is the first side to move. Therefore he must do everything he can to get his queen out early into the game. Moving the pawn in front of the king is essential for two reasons: 1) It creates a barrier in front of the king to evade checks. 2) It gives room for the queen to move quickly into the center of the board. The same goes for yellow.
Here blue and green must move the pawn in front of their queens. If they don't, then there is a small trick red and yellow can play. If blue moves the pawn in front of the king, yellow can check blue with the queen. No matter what move green plays, it's all over. Red will move their bishop diagonally to yellow's queen and its game over. Blue is mated and they lose the game.
Therefore it is a good idea if you want to stay in the game to move the pawn in front of the queen if red moves the pawn in front of the king first. This idea will switch however if red chooses to move the pawn in front of the queen first. The same tactics stated above still apply.
Next, GET YOUR QUEENS OUT!
Here red and yellow both get their queens out. Blue and green must follow suit and move the queens in front of their kings. This is CRITICAL for blue and green. This is because it prevents discovered checks that red and yellow can capitalize on knowing that you must defend the king.
In another variation shown below, red will have to waste a move by moving a pawn to defend against a pawn capture by a queen if blue chooses to move the pawn in front of the queen side bishop. Red cannot move their queen out on the board immediately.
Blue can then move their queen to the newly opened square by their pawn push and attack the yellows pawn on i-13, (the pawn infront of yellows queenside bishop). Yellow will be able to move their king side bishop to the pawn infront of the queen to attack blues queen. Shown below.
This is just one of the variations.
NEXT make sure to avoid all discovered checks. Red and yellow have two choices. Either move their king side bishop in front of their king to avoid checks, or move their queen side knight.
We chose to move our bishops.
Blue and green MUST move their knights to avoid discovered checks.
In the next pictures green fell for one of the simplest tricks in the book. Red was able to capture the knight and yellow checked forcing green to move. And easy way to stop this cute tactic is to move the pawn infront of greens queen up one square. This blocks yellows queen from checking greens queen.
By then it was pretty much all over.
Okay so that was just some examples of how your opening play in 4 player teams should look. Next time, keep some of these ideas in mind. Also, trust your teammate! The arrows are to show you SUGGESTIONS on moves you should play. Don't think of it as the teammate bossing you around/ telling you what to do (but most likely its a good move) Just communicate with him and ask what his idea is.
I hope this helps some, sorry for the horrible picture examples but I'm not a good graphics designer by any means. If you have questions or comments or other ideas you thought I missed feel free to post them below. Thanks!
-Reuben