Good rock shoes are the more important thing, initially ignore those that say you should get ones so small they hurt, get ones that are comfortably tight and get them from a rock shoe shop, tell them youre a beginner and they should help.
I’d just like to second this. If an “expert” in a shop tells you they should hurt, find another shop. They aren’t necessarily built for comfort, and people at the top of their game may want to have their toes crushed into odd positions for performance reasons, but as a beginner if your shoes aren’t comfortable then all it’ll do is discourage you from climbing.
I made this mistake. I bought a pair of the legendary 5.10 Anasazis after destroying my beginner pair and wanting something a bit more advanced. They’re the most amazing shoe ever but I bought the wrong size based on poor advice and they’re torture. I ended up getting another pair of better fitting, less technical ones for day-to-day playing and reserving the pink bastards for when I’m feeling adventurous. It’s probably not improved my climbing any, but it sure as hell increased my enjoyment of it.