What sound signature and other qualities of headphones do you prefer?
Everyone has a different taste: Some like bass heavy ones, others like heavy v-shapes and others still like something more well rounded. Additionally, there are other tonal qualities such as sound stage and imaging or even non-tonal ones such as comfort or even just looks.

Most of the popular open back Sennheiser headphones are pretty much the definition of a "well rounded" sound signature and have fairly tight sound stage with precise imaging.

Certainly not a bad pick but, for myself, I've found their sound stage to not be my preference for gaming; preferring larger sound stage even at the cost of accuracy. It might be totally different for you though.

Additionally, I'd recommend you to not skip on budget headphones until you really know what you like. I started out with 30€ Superlux 668b extra-cheapo headphones that I modded with 30€ velour pads and I've only looked into replacing them now 5-6 years later. What I'm getting myself now is certainly better but not quite as much of an improvement as you might expect.
Better buy something decent but not quite very good for little money initially and then buy something more premium that is truly fit for you a few years down the line when you know your taste better.

There's also nothing wrong with buying one of the rare decent "gaming" headphones when starting out. For most people, these are more than good enough.
When I started out years ago, I would have likely been perfectly happy with the O.G. HyperX Cloud gaming headphones which would have saved me a lot of trouble with the mic in the beginning. Don't sweat open vs. closed when it comes to sound stage; it's by far not the only determinator for sound stage; there are very wide closed cans and very tight open ones (past a certain point, open is of course better though).
(I don't know whether this particular model is still this decent or whether there is a better alternative these days; I haven't kept up with the !budgetaudiophile@lemmy.world market for years.)

You can still pursue upgrades after that; then get yourself a decent stand-alone mic setup and only then would I get proper standalone cans for "big money". I'd do it in this order because because mics and, more importantly, mic interfaces are a whole other rabbit hole to get lost in.

On the amp question: You will know when you need a dedicated headphone amp. If you don't know right now, you don't need one yet. You'll likely know after you've got yourself a decent mic setup.