Well if both bikes either have braze on front derailleur mounts (a bracket permanently attached to the frame) or both use clamp on derailleurs
and have the same diameter seat tubes then you can reuse the front derailleur.
A quick check with calipers or other measurement device will tell you if the seat post is the same size which will allow you to reuse the seat post. It's very likely the stems both fit a 1-1/8" steering tube if both bikes are of recent vintage but you should check to se if both stems clamp on to a similarly sized steering tube. If one or both bikes are older you should make sure they both use threadless or both use quill style stems.
You'll probably find it difficult to reuse all your shift and brake cables and possibly the housings depending on how each bike is configured for cable stops. Most bikes force you at the very least to disconnect the cables to the rear brake and both derailleurs to unthread the cables. Rethreading used cables with typical fraying can be difficult if not impossible. Sometimes you can use a high quality cable cutter (like the Park or Pedros tool) to cut a clean end on the cables and get them to thread properly but it's generally not worth the hassle as even the kink left from derailleur and brake clamping makes it tough to get them through the housings.
Depending on the mileage on your chain and whether it's a 7,8,9 or 10 speed chain it may or may not make sense to reuse the chain. Running worn chains speeds up the demise of cassettes, especially on the 9 and 10 speed versions. But if the mileage is low you can break and rebuild the chain onto the new bike but you'll need a chain breaker and possibly a new master pin or quick link depending on the chain style to put it back together again since chains thread through the frame and can't be moved without breaking them.
Also look closely at the bottom bracket configuration for each bike. If both frames are fairly modern and weren't built in Italy you can safely assume they're threaded the same. But Bottom Bracket styles may differ, tough to explain in a forum post but you'll want to establish whether the Bottom Bracket is reusable before you tear the working bike apart.
If these things aren't clear you should surf over to Park Tools or the late Sheldon Brown's sight for descriptions of these parts and instructions for removing, installing and adjusting various components.
Park Tool Website
Sheldon Brown-Bicycle Technical Information
-Dave