Basically whenever I shift into a new gear my exhaust is letting out a large smoke cloud that's visible enough for me to see in the side mirror. What's going in? Burning oil or something worse? Any help on this one would be great.

 

is the smoke blue? white?

My guess is valve guides/seals need replacing.
have you done a compression/leakdown test?

 

The smoke is blue I know that for sure. And no, none of those tests have been done. How expensive is this fix?

 

Depends on what is the problem. A compression check / leak down will help determine where the issue is stemming from.