If I want to get started on indoor climbing, what is the bare minimum I need to have, assuming I don't want to rent equipment?
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Shoes and a harness. Some places don't have belay devices or carabiners attached to each rope, so check on that beforehand, as you may need to provide your own.
Chalk and a chalk bag is something you'll probably want pretty quickly and often comes packaged with the harness.
For climbing shoes, look for something comfortable to wear for a couple hours but tight like a slipper. Imagine that you're going to be standing on your toes a lot, so you want a direct link from toe, to shoe, to wall. The same tightness should extend all around the toe box (i.e. gaps along your big toe or towards your smaller toes are bad).
For a harness, just aim for comfort - you'll spend a lot of time hanging on the rope early in your climbing career, so a bit of padding won't hurt you. Lots of harnesses have a ton of "gear loops", but for indoor you can get the simplest one possible, just somewhere to hang your chalk bag, if that. Padded leg loops are worthwhile though, as is a simple way to get in and out of it, and an obvious way to check that you are properly strapped in.
Belay device - hopefully they have a GriGri attached to each rope. If not, consider buying one if you can spare the cash. Otherwise, just get a simple ATC-style device and call it a day. In order to attach your belay device to your harness, you'll need a carabiner. Most people prefer medium-large pear-shaped locking carabiners for this purpose, but any large, D-shaped locking carabiner will do the job too.
As @xpda says, bouldering doesn't require much equipment at all. I found it much easier to start bouldering than climbing with a rope; I went a couple of times to a non-bouldering place and though I liked it, I didn't stay with it- bouldering was easier to get into and keep coming back.
As for the shoes- again, agreeing with @xpda - the first few times I went I found my lack of technique and underdeveloped muscles were far more important than whether I used climbing shoes or not. It was only after about a month that the shoes became a factor.
Of course, I started as an absolute beginner. If you're already experienced (or more talented) the shoes might be crucial earlier.