As others have mentioned, Winchester AAs are the gold standard for birdshot (just be sure you get a load that will run your gun if you've got a semi). Federal's Top Shot has been historically decent as well. I've not had issues with Rio in pump guns, but I trust that TC has. I just ran a case of Remington's heavy dove load in a one-day Fisher class, and had zero issues out of my 1301. My newly circumcised VEPR also likes it, so I'll likely be buying more. What you want to stay far far away from are the super-cheap shells like Winchester Universal, the kind that you buy in 100rd packs at Wally World. They tend to have very soft hulls that are prone to deformation, and cheap soft rims that cause extraction and ejection issues. I've seen more than one student show up for a class with that crap and spend the day mortaring every other round (and having to punch some out with a rod because they got stuck so badly that the extractor tore through the case rim) rather than learning. S&B buck and bird loads have consistently gone bang, but man do they stink of burning hair with the felt wads they use.
You'll have to ask the instructor for specifics, but for our classes, good birdshot is good birdshot (as long as it's lead), buckshot is split into "Verify pattern on your good/defensive stuff" and "this series of drills is a step up in recoil to confirm your push-pull technique, we don't really care how it patterns", and slugs are "confirm zero on your good/defensive stuff and use it in a timed/scored test". So, for our 2-day class that comes down to 400rds of reliable lead birdshot, 35rds buck (10 good, 25 whatever), and 15 slugs (you will want stuff that's got a good zero).