I recently had my clutch replaced.. pretty much the whole shabam. I've been told by a few people that I shouldn't worry about it.. the clutch needs to be "set in."
But, I keep getting the burnt clutch smell when I reverse and/or leaving in 1st/2nd from a cold start. When I got the car new, I never noticed this issue. Is this something that should go away after a small break in period? Or should I be worried and try to take it back to the shop?

I am pretty confident that I am handling the clutch correctly.. been driving manuals for 7+ years.. and never an issue till now.

I've tried searching and can't really find an answer to my question (maybe I suck at searching)

There is a break-in period but it should never smell like that. You have engagement issues.

Is that an actual problem with the clutch? Or should I have the transmission looked at, too?

Without being there or being the one who did the install it is hard to say, exactly, but I would get that car back to the shop that installed the clutch and not leave until the clutch no longer stinks. It is a clutch issue, though.