From your description it's likely that the pawl responsible for upshifting is seized and not engaging properly. The reason it would be able to shift the first two gears is because for the 1-2 shift the pawl face engages with an edge of the ratchet wheel instead of a tooth and since the trigger moves far enough that you can shift over two gears at once the 2-3 shift is possible using the same mechanism.
To fix this you would need to remove the cover and clean out the old grease using a solvent or degreaser, or even something like PB Blaster or WD40. You'll probably need to move the pawl back and forth to work the old grease out from around the shaft the pawl rotates on. Once the pawl is moving freely (ie, it snaps back against the teeth without hesitation) apply some fresh grease and put the cover back on.
This video describes the problem and fix, at about 1:08 you get a clearer shot of the upshift pawl and ratchet wheel. Most shimano rapidfire plus shifters use a variation of this design, even if you have a different model than depicted in the video you should still be able to clean and regrease the mechanism the same way.