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A lot depends on your own background. Do you already have a college degree?
If so, then there may be little advantage to earning another one
specifically in TC. A certificate course would work for you, or on-the-job
training.
If you don't have a college degree, then getting one (in any field) is
probably a good idea. Many of us have achieved a lot spending our time on
the job rather than in class (myself included before I ratted out the cause
and spent my nights getting a degree in english). But there is no question
you'll be eligible for more positions (particularly entry-level) with a
BA/BS on your resume.
Based on your post, I recommend taking a series of TW-focused classes or a
certificate program before hitting the job market. As most of us on the
list continually agree, knowledge of TW tools such as Word/Framemaker is the
least important part of being a good tech writer. Find a program that
emphasizes writing, editing, information presentation and project
management. If you're looking at the computer industry, also look for an
emphasis on instructional writing, audience/task analysis, usability
studying, etc.
A plug for my own school--University of Washington in Seattle.