Other sorts of objects can provide a new level of excitement. These are typically solar system objects (planets, asteroids and such), and earth orbiting objects such as earth launched satellites and earth orbiting debris. (There is always the moon, but it is mostly just a nuisance being so big and bright)
Asteroids, in particular, require up to date ephemerides. The edb files mentioned above have static information (Keplerian elements), that can only expected to be accurate for a few weeks. All bets are off if the asteroid in question passes close to a large body.
Good places to go for up to date coordinates are:
The Lowell Database (43082 orbits) (a 31 megabyte download) is 115 megabytes when downloaded and uncompressed. It is based on information from the Harvard Minor Planet Center, but orbital calculations are performed at Lowell. They update their data daily. Usually by 10:00 UT (3 AM MST), but on the date of the full moon, expect this to be about 4 hours later! Using the URL ftp://ftp.lowell.edu/pub/elgb/astorb.dat.gz is the most straightforward scheme here to get this data. The Minor Planet Center at Harvard CFA offers the MPC Orbit database for free download by anonymous ftp via the link: (MPCORB.ZIP is 22.1 megabytes and expands to 74 megabytes - as of late 2008).The Jet Propulsion Laboratory (JPL) at the California Institute of Technology has an extensive database of solar system objects. At the time of this writing they claim 430660 asteroids, 2966 comets, 168 planetary satellites, and more. They do not provide a database for download, but rather an interface to query their database and obtain information via several interfaces.
Earth orbiting satellites have orbits that are typically described by "two line element" TLE data files.
Celestrak is public. Space Track is restricted to approved and registered users (and holds 68,570,045 objects as of this writing in late 2008).