It really would help to know the specs of the MTB you want to use and also how far you want to take this, but here goes...
You could run drop bars. MTB bars are 25.4mm diameter, and road bars are 26.0mm, so you would need to ensure you had a stem with an appropriate bar clamp.
If the steerer on your fork is 1 1/8" and threadless, you should have no trouble tracking down a suitable stem. Otherwise you might be in for a bit of a search. Or if you have a threaded steerer because it's an old frame then you could get an adapter quill to run a 1 1/8" threadless stem.
The oversize 31.8mm standard bar diameter is used for MTB and road bars.
If your frame and fork supports disc brakes, Avid make a version of their cable disc for use with road levers. Alternatively you can use a travel agent as previously suggested (I think Delta make them, but I could be wrong there).
The rear spacing on a MTB is 135mm and 130mm for a road bike (being the standard for the last decade or more). A road hub will fit, but if this is a dedicated set up, you might like to build a road rim to a MTB hub.
Also, if you're using rim brakes on a MTb frame with a 700c wheel, the brakes won't line up with the rim. Mavic used to make an adapter to lift the brake calipers, but they might be hard to track down.
But a 700c rim on a disc hub (if your frame and fork has disc tabs) will solve that, or run a 26" rim with slicks.
A 10 speed cassette will fit a 9 speed MTB hub if you want to run 10 speed STI.
A big chainring like a 53 tooth will no doubt foul on the chainstay of the MTB unless it is spaced so far outboard that shifting and chainline go to hell, so get that idea out of your head,
Derailleur choice at the rear will depend on what gear set up you want to run. Front derailleur will be the MTB one, because as i said, road gearing is likely not to fit.
About the only thing that will fit without any clued in set up will be the saddle.
So I've either inspired you, informed you, or deterred you. If you want to be more specific about what specs your MTB has, or what you really want to achieve, we can tailor our advice and/or criticisms a bit better.
You say you don't know much about compatibility and building bikes - this project will have you becoming an expert in those little incompatibilities. Steep learning curve and all that.
Grumps