Typically, you can buy and sell stocks on your local exchanges via local banks or brokerage firms.

For the U.S. and other overseas stocks however, here are some info on the major brokerage firms out there and the markets they cover. 

Markets / Brokerages*TD Ameritrade*Interactive Brokers
*Saxo Capital*Tiger Brokers
Singapore (SGX)
Malaysia (Bursa)

U.S. (NYSE & Nasdaq)
HongKong (SEHK)

China(SSE & SZSE)
Australia (ASX & NSX)

* also available for Options Trading

If you would like to open an account with any of these brokerages, remember do perform your own due diligent e.g. check on the fees involved beforehand and get in touch with the respective company directly.