The choices for slides seem to be pots or linear encoders. For pots you need a system with many analog inputs, then you need to send them to your computer. USB seems like a no brainer. Might I suggest using a MIDI over USB descriptor. Many Digital Audio Workstation controllers work with this protocol, and the libraries for the PC side should be readily available.
In fact, if you're in a rush, you might look into the commercial availability of these controllers. In the Audio Workstation world, they tend to call them "control surfaces"
"Cheap" is relative. About $300 gets you this, or plenty of others like it, which might have enough sliders and pots to meet your needs, and you can start to program the host side right away.
You also know it will work, and you don't have to worry about design. This is a huge advantage (unless, of course, you're doing this for the design experience!)
I don't think you'll be able to build anything like what you describe for much less than $250-- assuming you own everything in your development chain already, so we're in the ballpark.
Pots would probably be the less expensive option, and probably easier to program, but you need to make sure that you get a microcontroller that can handle the job, or a separate a/d converter.