At some point in your practice of Olympic Weightlifting, you will probably want to consider buying a pair of weightlifting shoes. Unlike running shoes, weightlifting shoes have a hard sole designed to be stable under extreme loads and they provide a little heel lift to facilitate an upright torso when in the various squat positions. The hard sole also efficiently transmits force whereas a running shoe absorbs force. Bud Charniga, who runs dynamic-eleiko, has written a great article on why
we have weight lifting shoes.
So where to buy? No athletic shoe store or sporting goods store carries them so you will have to order them online. Best to get a current measurement of your foot, US size and International equivalent.
Here's some options: www.dynamic-eleiko.com carries a full line products for all your weightlifting needs, including Adidas shoes. Adidas is far and above the most popular brand of weightlifting shoes out there. You pay more for them, but they are the best.
werksanusa.com carries Werk-San shoes, Wan Hoa (the Chinese Team shoe) as well as a full line of other weightlifting equipment. These shoes are cheaper than Adidas, many lifters like them.
http://www.roguefitness.com carries Do-Win Chinese shoes. Again, less expensive than the Adidas shoes, and many lifters really like them.
Finally, Ristosports.com carries Risto shoes which can be customized and they also carry the shoe designed by the great Bulgarian weightlifter Stefan Botev.
VS Athletics has a shoe also...I don't know much about these except that they resemble the Chinese shoes in design.
The selection and availability of weightlifting shoes is becoming greater every day. These are the ones I'm aware of at the moment. Please feel free to add any other suggestions in the comments. So, shop around, find a style you like in the price range you can afford and buy a pair. Good shoes are maybe the second most important piece of weightlifting equipment after a good bar.