The practical maximum speed of G mode wireless is about 19Mbit/s (
ref Wiki) but in practice that is achieveable if you have no overlapping networks within range and you are operating in 'pure' G mode. A lot of routers default to mixed B/G mode, to accomodate older cards, so check the router setup to make sure your is in G Only mode. You can also try changing wireless channels to see if you pick a less congested one (a lot default to 7 or 8, so try picking 1 or 13). If you want see what other networks are in the area some wireless adapters have a utility to scan the area, or check out Kismet.
Having said all that, yes, N is more likely to make full use of your bandwidth, providing all adapters and router are upgraded.