For indoor climbing (oxymoron?) you should start with your own shoes. Brands like 5.10, La Sportiva, Boreal, MadRock all make good shoes. Get shoes that fit the tight without pain. Try on as many models as possible.

Next you need a chalk bag; pick you favorite color or pattern.
Nothing technical here.

You now need a harness. Black Diamond, Metolius and Petzl make good ones. Again try a buch on.

To go with your harness you will need a locking carabiner designed to be used with a belay device. For belay devices I like the Black Diamiond ATC. Petzl also makes the GriGri which is a bit more complicated but is a good choice for indoor and sport routes.

That is the basic gear you need for climbing in a gym.

Don't stay inside; use the gym to propel you into outdoor climbing on real rock. Redal rock can't be beat.
Bill