The big advantage pump shotguns have (compared to equivalent quality semiautos) is price. While inflation has mostly put an end to the days I could pick up a good used 870 Express for under $175, a good used pumpgun is still going to be less expensive than a good used semiauto, and that's a definite given for new guns as well.
As I've said before, reliability should be pretty much equivalent, given well maintained guns and good quality ammo. The pump of course requires a trained and practiced operator to avoid short stroking ... but in my not at all humble opinion, ANY defensive shotgun requires a trained and practiced operator to be reliably effective and safe in use. Shotguns are at the same time the most simple and the most demanding of all defensive firearms IMHO, but I still prefer them hands down for any short range short duration defensive need. For as trainer Clint Smith says, shotguns will remove meat and bone. In short IMO there is nothing as likely to prove effective in self defense against a human assailant than a shotgun in the hands of a capable shooter.
With that said, the hardware matters less than the shooter. Even before I retired, it was an accepted truth in US Army special operations that 'Humans are more important than hardware.' And COL Boyd said it even before then - see my sig below.
So as long as the shotgun is mechanically reliable, the choice is more a matter of what fits and feels right than much of anything else. There are lots more good shotguns out there than there are good shotgun shooters.
I'm an unabashed 870 fan, but Big Green is not the be all and end all. I would not feel at all under-equipped with a Mossberg 590 either. For that matter any solid design since J.M. Browning's Winchester 1897 would do. The thing is, I've been using 870s so much for so long that my hands know the design like my tongue knows my teeth. I don't have to think to run an 870. So my advice is to pick a design, whatever you like, and get good with it - then stick to it. Marry it, in other words.
I have a bevy of pumps - a Norinco clone of the solid frame 1897 riot gun, JMB's next repeating shotgun design, the humpback Stevens 520 (which imho has the best take-down system ever), the mechanically similar but streamlined Stevens 620, a police trade-in High Standard Flite King riot gun, etc. Any of them will do. But my hands just don't know any of them as they do an 870.
As to the class, get your safety chops and administrative gunhandling chops down pat first, before you take a class. Safety is primary - the Four Rules should be reflex. Muzzle awareness and trigger finger discipline should be absolute. Being comfortable loading, unloading, shooting and cycling the gun should be a given. When you're comfortable with the above, you're ready for a class.
I'd encourage you to go for the best training you can find. While there are a lot of local folks who can handle shotgun instruction quite well, try looking at a top tier nationally/internationally known instructor. Unfortunately we just lost the person many considered the best in the business with a shotgun, Louis Awerbuck. But the state of the art lives on, in very capable hands. Consider Randy Cain, John Farnam, Clint Smith, Tom Givens ... several of them travel to some degree, though none as much as Louis did. It's possible you might be able to find a class close by you, brought by one of the itinerant instructors, or you may well find someone local you like. My advice is to sample several classes and use what works best for you.
Good luck and happy shopping...