A decent disc system is quite expensive - you need disc ready hubs, disc ready fork+frame and the actual disc brakes. A cheap/medium cost set of rim brakes (V-brakes) will typically perform as well if not better than cheaper disc brakes (and the maintenance is not hard). Thus, budget minded people should avoid disc brakes.
As for building a bike piece by piece, you'll typically end up spending more than buying a bike whole. Also, if you don't have experience with bicycles, you won't know which parts will fit together to give a suitable geometry for you (thus, buying a pre-built bike is a good idea) -- you need to know about standover, effective top tube length, all the angles involved, the Q factor you get with the choice of bb+cranks, etc which is hard to spec without experience or having a bike which is similar to what you intend to buy. Generally though, at the end of the day, you're going to pick parts based on your measurements and goals and then matching standards for each part. Park tool and Sheldon Brown are good places to start for looking on how to build a bike. Note that assembling a bike also requires some special tools (cassette tools, etc.), so you will be spending money there too. Thus, I do not recommend you build a bike part by part. (*)
If you're on a budget, you're better off buying a used bike (plus, you can test the fit and everything). For high slopes, you want low gearing (big cogs in the back and not too big things in the front). A used mountain bike with slick tires may be a good option (you don't need suspension on the road either, and this would make a good commuter and be cheaper than most road bikes -- think 80s Specialized Hardrock or similar). Also, your slope estimate is probably quite a bit off (20 degrees is quite extreme and rare to begin with!).
(*) If you live near a bike co-op, they might be able to give you access to the tools and people who can help you there, but it will take time and generally requires some experience anyway. Thus, I still stand by the idea that you should buy a used bike.