Hi, welcome to the forums.
You packed a lot of “easy” questions in a short opening post. The answers can be found in these forums and over at the blog.
Search is your friend.
Also, we like questions as concise as possible. That way we can be of most assistance.
Fundamental parts of DJ-ing. Lots of stuff’s been written on this one. But in a nutshell my opinion on the matter:
1) What comes next. As my co-moderator Terry wrote on the blog (be sure to read his article!), knowing what comes next is the apex skill of any DJ. However, this will only come from LOTS of practice in playing out. It’s a skill not teachable (coachable maybe) and not practicable in a bedroom setting. Before going out to play you will have to learn a few other skills.
2) General musical skills. Like knowing about keys (for harmonic mixing), song structure (to know where possible mix in and mix out points are) and things like beats, phrases, downbeat, droppoints, acapella’s, instrumentals. And of course COUNTING and the way DJs count (1-2-3-4, 2-2-3-4, 3-2-3-4, 4-2-3-4 … 8-2-3-4, 1-2-3-4 repeat), which you need to be able to understand sync and manual beatmatching.
3) General technical DJ skills. This is highly dependant on the kind of DJ you want to be. Clearly if you want to be a scratch DJ, you’ll need to learn how to scratch. Otherwise, it’s a wishlist kind of skill at best. Things you will need to know are how to cue up a track, how to use cue points, loops, FX, how to nudge, manual beatmatch, change pitch, just to name a few. Get Phil’s How To Digital DJ Fast course and you will have a good idea of what to start practicing and – as important – how to practice it properly.
4) General software/controller skills. Once you know what to do and what to practice, it pays to know how your software and controller help you do those things.
As for music. Plenty of tips in previous posts and blog articles, so not gonna rehash those.
We currently recommend a few beginner controllers:
Denon MC2000
Numark Mixtrack Pro (2 or 3)
Reloop Beatmix 2
Pioneer DDJ-SB
Which one to pick is highly personal and also depending on your (intended) workflow, choice of software and type of DJ-ing you intend to do. Again, go through the forums and blog to find pertinent information. Plenty of it around.
Hope that helps some.