join a club, get to know people, find out about courses and clinics, do those, get mad experience and progress...
in the mean time for indoor climbing and bouldering, get the following gear:
*climbing shoes, easily the most important part, they should fit snug but shouldn't hurt your feet....
*a harness, go for one with decent padding and easy adjustability ....
*a belay device+hms carabiner, i personally would suggest a manual belay device such as the black diamond ATC guide, this is a good device that will also serve you well as you progress and start climbing outdoor or doing alpine stuff.... the automatic devices such as a grigri are cool and all, but i feel as if they give you a wrong feeling of safety and they're a bit more difficult to handle properly (giving and taking rope can be much more difficult with these devices..
*a chalkbag (optional, i almost never use mine, except for indoor)
*a crashpad, this is basically a foldable mattress to absorb your impact when bouldering outdoor
now i suggest you join a climbing club and have those people help you out further, they will teach you how to do sportclimbs as a follow and as a lead climber ...
additional gear once you've got some experience and start to go outside...
*quickdraws (+- 7), but totally depends on your region, in belgium we usually have enough with 7 quickdraws for each route, but this can vary from place to place
*prussik cords
*additional hms carabiners
*a belay sling
*your own rope, (i prefer 75m , but it depends on your home region)
*a good ropebag, decent rope bags have a built in tarp you lay on the ground that has 2 loops, 1 loop has the end of the rope, the other the beginning, so you basically belay on top of the tarp, when you're done climbing the rope lies on the tarp, you simply tie the end to the free loop, fold the tarp with the rope on it together and put it in the bag. next time you need it you simply unpack and unfold the tarp and you're ready to go! no more coiling trouble 
 
^^this is about most of what you need for sportsclimbing on equiped routes
if you want to progress and start climbing unequiped routes you're going to need trad gear, but i'm not going to school you on that, if you have to ask what trad gear you need, you're not yet ready for it ...