New Skater: should I buy my own skates, what kind? Or rent

So I’m a brand new figure skater. I’ve skated through out my childhood so I’m comfortable on the ice and know how to balance, skate forward, backwards, swizzle, glide, but it’s my first venture into figure skating so, I’m not sure if I should just rent skates for my first lesson and ask my coach about it later, or just start off with my own pair. The problem with that though is I don’t want to buy skates that are perfect only to have to buy new ones a month later if I do happen to progress quickly or find out I’m at a higher level that previously thought (or a lower one, you never know). So with that, any advice? Any beginner skates that have a nice transitional aspect to them? Should I wait it out?