I did this on an old Trek for my wife. Since you have matching cogs, most of your work is done. Sounds like (as long as your rear derailleur) has the same throw/ratio as your shifters, you just need to install, run cables & adjust. Don't forget new bar tape. Brakes will work, but how well will depend on the compatibility (wait and see). I moved an entire group over to my wife's Trek, that way ensuring everything was a match. Let us know how you make out.