When you got the bike last year did you ride it at all or did you just store it? If you rode it, how was it then?
Although it appears you do have issues with the carbs. I wouldn't start there. The maintenance history of the bike appears to be unknown therefore I would first check the valves and ignition timing.
First check / adjust valves and Check ignition timing
If all is good there then focus on the carbs
Read the Troubleshooting Section (Section 20) - Poor Performance at Low and Idle Speeds
http://dc384.4shared.com/download/_Y3eGfeo/Honda_CBR_600_F2_91-94_Service.pdf
Valves with incorrect clearance can cause the symptoms you are seeing.
Also, syncing the carbs when the valves are out of spec is a waste of time.
If you are not mechanically inclined bring the bike to a mechanic or someone who is. Cost will mostly be labour if there are no major issues, and will vary between shops. ** Make sure to get a quote **
Valve clearance in the manual (link above) is on page 3-5
Valve Adjustment Guide- CBR 600 F2/F3
http://cbrforum.com/forum/stickies-91/valve-adjustment-guide-101969/