Unless you have the technical prowess, the time, and yes...the money (as it would still be expensive) to build your own tracking mount, you probably won't be able to do long-exposure astrophotography.
You can avoid star trails, and do "short-exposure astrophotography" by using three things: Very short focal lengths, very fast lenses, and very high ISO. You should be able to expose up to around 25, maybe 30 seconds before you get noticeable star-trailing.
The wider the lens, the less star-trailing you'll get, since the motion of the stars across the sky accounts for less sensor space the wider you get. A faster lens, obviously, will help you expose a brighter image in less time. CA and other optical aberrations can really kill detail on lenses used wide open, so higher quality lenses that correct for optical aberration will be a huge plus.
Finally, a camera with good high-ISO low-noise performance will be essential. A full-frame sensor is a big plus here, and the Canon 5D II is a big favorite of a lot of CMOS astrophotographers. ISO 1600 an ISO 3200 are not unheard of when doing short-exposure astrophotography. Just be wary of noise reduction in post...stars are true point light sources, and noise reduction algorithms can eliminate stars along with noise if you are not careful.
Depending on your goals, you may be able to get away with higher ISO than you might otherwise. If you are taking still shots, there is only so much you can do before you start losing useful detail. If you are shooting time-lapse, then you can probably get away with really high ISO without it being a huge problem, as the size and detail necessary for video is a lot lower than it is for stills.
If you really want to get crisp, clear, saturated stills of the night sky, then you really need a tracking mount. The cheaper Altazimuth mounts will get you stable tracking for shorter periods of time, however the more expensive equatorial mounts are really necessary to get stable, lengthy tracking at the earths surface. Any exposures or stacked exposure sequences that span an hour or more will probably need an equatorial tracking mount.