Below are all personal opinions and points. I'm still somewhat amateur myself but have been in the studio for a few years now.
1) What to you guys usually do when you begin a new project either personal, or with another artist?
With personal projects, I usually start out with some sort of skeleton of a song or album. I haven't recorded an entire album yet, just random songs. So, with songs, I start out with chords and vocals. Sometimes I might start out with screwing around with drums. Personal songs could start any which way. My best advice here is to play to your personal strengths and find a workflow that works best.
When it comes to recording others, it is a lot different. I start by recording scratch takes with a metronome (usually a vocal and a guitar, vocal and a piano, anything that the drummer can follow). Then, I get the drummer in, get the kit set up (new heads, tune it, etc), and mike it up. Get the drummer nice and warmed up so that they can play with the click and scratch track that you recorded earlier. As long as the drummer isn't drunk, hungover, or stoned, give them a set of headphones, run the scratch through it, and record.
Some drummers can't play with a click. In this situation, I will record the entire band "live" but only keep the drums. Sometimes I'll keep the other tracks if they were played well. This is less than ideal because if the band already has issue playing together, it will be very (!!!) difficult for them to play back with the drums. To avoid this mess, tell the band to go home and practice, or better yet, hire a drummer who play in time. I have a small arsenal of studio musicians I can use in events like that. The musicians/bands don't always like it, but this could make or break the record/EP/song/whatever.
2) When you actually begin the recording process, do you try to get the best recording possible and edit as little as possible, or do you like to fix everything in post?
I prefer to get the best recording possible. For me, it sort of depends on the instrument.
"Best recording possible". Especially with drums. Drums are a beast to edit. If the drummer can't play, sometimes there is no way to save a recording. The guy needs to go practice or you need to hire a studio drummer. If the drummer is even somewhat on pointe, you can fix the stray hits/misses with samples or even a slight nudging of the waveform without compromising the groove or beat.
When it comes to the rest of the instruments (horns, guitars, keys, etc.), it is very easy to just delete a poor take and record a new one. Sometimes guitar guys will have a hard time playing specific parts of songs

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, so in that case, get him to play everything he can with ease, then go and punch in the hard stuff.
If you are recording jazzers, they'll play everything well on pointe and together. Usually the best situation here is just to mic everyone up and let them play live. Usually makes for the best atmosphere.
Anyways, the point I'm trying to make here is that it is important to get the best recording possible to save yourself lots of editing work down the line. I like to save post production work for "gee whiz" type of aspects of a song (drop outs, automating FX, etc)
3) What does the mastering process involve?
I won't answer this question because I have very little experience in this realm. The
Wikipedia article on this topic is pretty well informative and should give you a 30,000 foot view into what it is.