GPS signal does not require Wifi, but navigation kind of does. There were some GPS signal conneciton issues early on, so you'll want to figure out if you're having those issues. If you are, we can help you sort them out, as Google has released updates that include a fix. That said, GPS just finds coordinates, so if the map isn't available offline, then it'll just give you a blue dot without a map, which is meaningless.
You can set a map area to be available offline. The method abbott mentioned should work, but they've also added it in a more logical place. With the map displaying what you want available offline, tap on the search bar. Some info should fill the screen. Scroll to the bottom, and there should be a spot that says "Make this map area available offline". It does have a limit to the size you can make available offline at a time, so you can't do a whole state at a time, for example.
Changes in navigation also require an internet connection. You can get directions before you leave, and as long as you stick to the route, they should work, but it can't recalculate the route without the internet.
There are some apps that are made specifically to allow for better offline navigation. I've started playing with NavFree, which lets you download a state at a time worth of maps. I haven't really used it much, though, so I can't say how well it does. For the price (free), it may be worth a shot if you need to do more offline navigation. There are also some paid apps that probably give a richer experience.