It's impossible to try and write a guide on how to handle each and every threat in the Battle Royale since there's hundreds upon hundreds of pre-set Pokemon the NPCs you face off against will use, but here's a healthy collection of tips to help you win as many battles as you can.
Double check that you can cover for weaknesses
Only having three Pokemon cuts in half your ability to have some sort of defense, so make sure you never have more than two Pokemon sharing a common type weakness, or else you'll end up with a team that has an utterly crippling weakness!
Check and double check time and time again that you can redirect a poor matchup to someone else, and that your lead is capable of handling as many threats at once to mitigate the question of switching at all.
And that's because...
Switching is usually a very bad idea
With you versus three others, the chance of you getting ganged up on by two or three trainers at once is high and should always be considered (and feared).
Nothing sucks more than switching out, and getting pummeled, and fainting before you ever had a chance. It is of crucial importance, now more than ever, that your lead Pokemon be as capable of a soloist that it can be.
Slow Pokemon could potentially cost you the game
Bringing durable, tanky, defensive Pokemon to sponge hits sounds good in theory, but the number of KOs you amass matters, and defensive, slow Pokemon usually aren't so good at that.
Worse, if you have a slow tank, there's a high chance an NPC will KO the Pokemon you were targeting before you get to move.
If you attempt to hit a Pokemon that gets KOed before your chance, the move will fail instead of get redirected! Yes, you read that right. As a result, Speed and priority moves are an absolute must.
But don't go too overboard...
Most fast and powerful Pokemon are also very frail
You need to balance your team properly such that you have Pokemon who are defensive and Pokemon who are offensive. Bringing all of only one of the two kinds can become unbalanced and makes it more difficult to pull the wins you want.
If Speed is so important, it makes sense to bring fast glass cannons to sweep the floor, but remember, it's not a one versus one match, it's a four-trainer free-for-all, and you'll have multiple pairs of eyes on you to lose!
A diverse team with tanks and speedsters will help you accomodate a variety of threats and adapt easily as the match progresses.