Hey man. This is cool. Exciting stuff. First question....are you in or near a major city? If so, it's going to be easier to find a studio that can deliver the goods so to speak. Typically it can go like this......you find a studio that has recorded/mixed songs/albums that sound good to you and you contact them. Tell them a little about your project and then go to the studio for a quick tour. Meet the engineer(s). This would be the one key "fancy person" without whom you will have no recording.
Generally speaking, a good artist or band who are well rehearsed and have thought things through can book studio time at a good facility with a capable engineer at the helm and walk out with a great sounding product. This would be assuming that the same engineer who was recording you (miking everything, pressing record etc.) would be the same one mixing the song. Some people like to have different sets of ears on the project so they will have one person record the session and another mix it. This wouldn't make sense for one song because it would be more efficient for the engineer to "mix as you go". In other words, instead of just listening back to raw tracks as you layer and overdub on top of them, the engineer would sort of be cleaning things up, tweaking and EQing during the process of overdubbing. This is a cool way to work.
It's actually what I aspire to do when I get back in the studio. Singles used to be recorded this way all the time. A song in a day. There's something really great about just focusing on one thing and not leaving decisions on the back burner. That's what kills recording projects. Example- We don't know what sound we want for this guitar or that guitar, or how we want to pan them, so let's just record them dry as a bone and make a decision later..........as opposed to.........we all have ears, and we're committed to tracking and mixing this thing right now, so if there's a decision to be made, let's make it. What sounds good? Analog delay? Great let's track that guitar with a delay pedal so the guitarist can be inspired by the sonic quality of it.
It's a more live kind of an approach even if you are overdubbing one thing at a time, brick by brick which it sounds like you would be. And you could do it over the course of a few days....don't let that song in a day thing scare you. I was talking about the early days....fifties, sixties etc. ....4 or 8 tracks....monster session players. They'd just bang it out and mix it on the spot.
As far as mastering.....you probably would get a third party to do this. This is different than mixing. Mixing is taking all of the tracks..... 16, 24, 48 or however many tracks you have (a drum set can easily account for 10 tracks with all the mikes people put on them. That's not my style but whatever) and mixing them down to 2 tracks...left and right.
Mastering is taking that 2 track stereo mix and making it "pop" which is what you want. That's probably all you should know about the mastering process. Let it remain mysterious. A good studio could tell you whether they have a mastering engineer they work with or whether they would do it there. Mastering one song might be worth just doing at the same facility. It would only take a fraction of the time that tracking and mixing would take so don't be scared off by the mastering process. It's the final step.
The other "fancy person" that you might want to strongly consider is a producer. If cost isn't too much of an issue and you're really gunning for a stellar result AND you have no recording experience it might be something to think about. It's real easy to get bogged down in minutia during the course of a tracking session and at the same time miss some other crucial detail. A producer has an objectivity that an artist ( especially a singer ) often can't have. Many engineers are comfortable wearing the producer hat for a session because they do it so often by default.
A lot of people think a producer is the person who puts their sonic imprint on something and often that's true. But he/she can also just be the person who sort of moves the session along, keeps the momentum going, keeps the energy up, listens for mundane stuff like a stray snare hit so the singer/ songwriter doesn't have to. Performers need to be able to be in performing mode on tracking day. And professionals need to be there to capture the great performance when it happens.
You could also ask a musician friend to produce the session for you. But it would have to be someone who had their act together and had a fair amount of session experience.