That's actually a whole set of questions on the subject of combining GPS with other services.
Most of this stuff actually goes beyond using the Android OS and extends into databases and online services.
Can I get the GPS coordinates of nearby phones?
To answer your first question: Yes, you can track GPS locations of phones. The big thing with this is that you need the phones to push their GPS coordinates to a server. The next step is then to ask the server, using your own GPS coordinates as the point of interest, what other GPS coordinates have been recorded in, let's say 200m. However, I must point out that this only works when you build an app for it and ask people to install it and run it. Also, there's a bit of a privacy issue with it as it's not really a mainstream thing to ask people for their GPS locations for no good reason let alone store that information on a server.
Can I create a personal navigation app like Tom Tom?
Yes, of course you can create a personal navigation app. There's open street maps that feature a lot of the worlds mapping data for free. Check out their website for more info: http://www.openstreetmap.org/
Can I retrieve the location of nearby shops, banks and other venues?
Yes that's possible, you can retrieve your current location in Android and combine that with a search engine that supports GPS. The first step in the process is getting the user location. There's a pretty good API guide on how to determine the user's location over on Google. Sadly, noone has yet written a blogpost (at least not that I've found) about it specifically for Mono for Android. It doesn't really matter though, as the java code is easy to convert into C#. Check the API guide here: http://developer.android.com/guide/topics/location/strategies.html
You can combine this piece of tech with services like the google maps API and query for adresses near a specific location. There's a few options here ( Given GPS coordinates, how do I find nearby landmarks or points-of-interest? ), I personally really like asking four square for venues near a location, because that one is actively maintained by a huge croud. Makes for a much more up to date experience.