What's the best way to increase the range of my 802.11g router?

"The best way is to buy an extender [repeater], but those are expensive.
Buying a better antenna (if you have a weak one) is slightly cheaper and can help as well.
But not all routers make it easy to change the antenna, in some cases they glue it in even though it's connected as a screw-attachment.

Other than that, trying to optimise the location of the router might help.
High and in the center of the area is usually good.
Some antennas come with an extended cable so you can locate them in a better place even if the router is wall-bound."

freeantennas.com has a couple of templates for using cardboard and aluminium foil to make your wifi more directional and increase the range, unfortunately the site seems to be down today so I can't send you exact links. If you google for windsurfer templates you can find them.