A GFCI (properly wired) will protect all the "downstream" outlets in the circuit.
Yes, you could tap into any accessible 120V circuit, assuming you follow proper wiring protocol and procedures. Many GFCIs are rated for 20A circuits too, by the way, if that's what you have—consult the packaging to be sure.
You can also get GFCI breakers to replace the breakers in your power center. I don't know offhand if tandem GFCI breakers are available (it may vary depending on the circuit breaker family used). If you can make use of a GFCI breaker, it avoids the need of putting in an electrical box etc. for the GFCI outlet.