I haven't played USUM or SM before, but I will try giving an answer. Here goes! :D
1. The Right Pokemon
So you will need a Pokemon first of all, that is higher levelled or similar level to the Tapus you are trying to catch. Good defenses aren't a nessecary, but bringing a Pokemon that resists the Tapus moves will help a bit better, so that they don't KO you when throuwing Pokeballs. Tapu Bulu is a Grass and Fairy, so a Poison, Fire, or Steel type might be good to resist. Tapu Lele is a Psychic and Fairy, so a Steel type might be good.
Moves that you will probably need are False Swipe, to get the opponent to 1HP, and Thunder Wave. Often moves with a chance of inflicting a status are risky, and Burns/Poisons/Bad Poisons can faint the Tapu if you don't catch it in time. Another thing to note, particularly for Tapu Bulu, is that they are Terrain setters. Tapu Bulu's Grassy Terrain will heal overtime, so you might want to change it to something else so it doesn't heal the False Swipe damage.
First, no matter what Pokemon it is that I am catching, Legendary or not, I always try to throw a Quick Ball first. I caught Galarian Zapdos and Regieleki with one first try, so you never know! Should that fail, the next thing to do is use your Pokemon to whittle the Tapu down. Ultra Balls are good for this after it's low, and then Timer Balls are your next best bet. If you feel it's close to Stuggling, Timer Balls will save your butt. Other balls of note are Dusk Ball at night, and Heal Ball because it might just work.
Good luck on catching the Tapus, hope I helped!!