"One thing that is sometimes possible with pawns that are close to promote is something I want to call mutual promotion. Normally 2 pawns in the centre with different colours will stop each other. However, sometimes it is possible for a player to advance a pawn and the other player can choose to also advance his pawn instead of capturing. This way both players choose to promote pawns instead of stopping each other. This can be seen as among the best results both players can hope for. Keep in mind though, that especially at lower levels the other player might not understand this strategy, the pawns get traded and the results is bad for both of you, so there is a risk with using this strategy."
Indeed, and if you are the 4th player to act, you basically have no chance to play for promotion (or stop it) if the two players before and after you do so (since a queen pawn move will allow a bishop trade, and a king pawn move a queen trade). Getting a queen is not winning at all, but sometimes the center gets blocked and the two remaining players don't have a chance to promote later, when the extra material is evident. I think a good strategy then is just to make a long pawn chain, defend the base points and basically wait as long as you can, while defending threats from either side. Attacking would be suicidal.
"Overall it is very solid to fianchetto. However if your opponent fianchetto on a side first, then you typically do not want to fianchetto on that side since you do not want to trade bishops (remember, trades are bad for you)."
I disagree with this one, I think it's still beneficial to fianchetto, but you must play the move corresponding to a3 or h3 on a standard chess board, something like d3 and k3 in 4-player-chess. In some sense, if all players were very good, offering a fianchetto is not a problem, since the other player would not benefit from trading. If someone offers a fianchetto back, I always decline the trade with d3 or k3 (or other corresponding blocking moves).
"When it comes to castling, one interesting advice is to always castle to your right regardless if it is kingside or queenside."
This one is very much a dynamic decision (DELAY castling as long as you can, especially if noone can check you). If the player to the left does not fianchetto against you, and/ or has traded something after 10 moves, you might want to go left. Sometimes a sneak attack against the left player might actually be very effective. Once I was in a bit of trouble in the center and went for mate against the left player, as it turned out this all out attack turned everything around, since the other players suddenly had free pieces to take from this player, while I was attacking a king that only had a bishop defending it.
If I had to explain the most basic strategy for this variant it is funnily enough very similar to what I would say to someone that is learning chess. Make sure your pieces are safe and capture when it is free. Also keep in mind the beginner tips I posted here.
It may sound very simple, but if you are rated below 1400, then you really do not need to think too hard on other strategical stuff and focus on board vision and tactics. Besides, if you make an elaborate attack against someone and that player is forced to respond, then that is not necessarily a good thing for you. Other players may take advantage of your attack and gain more points than you do or focus on something else like promoting pawns.
The next thing I want to focus on is pawn promotions. If you have played some games you have probably realized that it is relatively easy to promote pawns since the 8th rank is in the centre. Promoted pawns are also only worth 1 point to capture. The first advice I want to give regarding promotions is that if a player is getting close to promote, you should not panic. If you end up sacrificing a bishop for a promoted pawn, then that is not good for you since you only gain 1 point. So if you are in a position with 4 players left you can just let a player have his promoted pawn and it is not such a big deal for you. It takes a lot of moves to promote a pawn, so that player might be a bit behind in development regardless, so even if he has an extra queen that might not be a big threat to you. It is a bigger deal when someone is developed and has 2 pawns about to promote and wants to get 2 queens back to back.
While you should not panic when it comes to other players promoting it is still in your interest to stop it if you can do so in a convenient way. The most ideal for you is if someone else deal with it, but that is not always the case. The main way is to prevent it with a pawn or create a threat against the player that is trying to promote to give other players more time to deal with his pawn. One thing NOT to do is create threats against the player that you want to prevent the promotion. Stopping him from stopping a player to promote is among the stupidest things you can do, so if you have no clear benefit like getting free points you really should let him be for at least a few moves.
One thing that is sometimes possible with pawns that are close to promote is something I want to call mutual promotion. Normally 2 pawns in the centre with different colours will stop each other. However, sometimes it is possible for a player to advance a pawn and the other player can choose to also advance his pawn instead of capturing. This is the most common to do with the player in front of you with both players moving their pawn in front of their queen. This way both players choose to promote pawns instead of stopping each other. This can be seen as among the best results both players can hope for. Keep in mind though, that especially at lower levels the other player might not understand this strategy, the pawns get traded and the results is bad for both of you, so there is a risk with using this strategy.
When it comes to king safety and castling, one interesting advice is to castle to your right regardless if it is kingside or queenside. The idea is that you want to stay as far away from the player to your left as possible (your worst enemy). There is no general rule where your king is safe though. Sometimes castling left is better and sometimes you might want to just walk your king to the side and keep your rook in the corner (especially true if your rook pawn is gone). It can often be a good idea to delay castling since it is a very dynamic decision where your king is safe, but of course it is better to castle too soon rather than too late.