How should an amateur go about setting up a home-studio on a budget? Specifically sound absorption and sound proofing?Well Patrick. That's a fantastically easy question to ask, which has a phenomenally complicated answer, so I'm going to have to pick on you a little.
First up, one of the easiest ways to have someone like me dismiss your question is the phrase "on a budget". At best this usually means "as cheap as possible", but it's a coy way for the pressure of deciding on a purchasing decision to be dropped on us advice givers. Now we need to guess what "on a budget" might mean to you. I elaborate on this more in a previous article: Stop Spending As Little As You Can, Start Buying As Much As You Can Afford
So after reading that article, if you're "on a budget" that means you must have a budget, and if you want helpful and accurate advice, don't be shy! Tell me your budget!
So to better discuss your question, you are correct. There are two components to treating a space: sound leaking into and out of your space, and sound bouncing around inside your space.
For sound leaking in and out (absorbing) you need mass. Pretty much the only way to fix that problem. Often this is something we simply can't do. If you can tear open the walls to fit higher quality insulation, if you can build a free standing structure full of insulation, then you can throw mass at the problem. Otherwise, you just need to find the quietest corner of your living space, and record around the sounds that do leak into your space. I don't record much during the day from home because my apartment is next door to a school for example.
The second half of the equation is even trickier. Treating a room for reflections is an art, and you'll need to learn to use your ears. Too much foam and other acoustic treatment will suck the highs and mids out of your recordings, too little will mean a "live" echo-y recording. In smaller spaces low frequency energy has a tendency to build up, and the only way to curb it is through mass (again) or microphone selection. The VOLUMES of information written on bass traps is staggering.
So my advice is always "Start Simple". Get a mic in your space and see how it sounds. Play with distance from the mic, play with angle of recording. Listen to a couple different adjustments, then assess what you specific problems might be. From there start planning a budget on what it will take to clean the impurities out of your recordings.
If you're lean on cash, you need to learn a little about sound reinforcement. It's not going to be good enough to pick info out of forums, or to continue getting snippets of advice. Go to the library. Learn a little about the physics of sound. You don't need to master these concepts, but if you can't afford to pay a professional to go in and tweak, you'll need to make it up with a little Sweat Equity. I wrote a little more about this in Combating The Casual Understanding of a Concept.
Lastly, if you've listened to the audio on any of my reviews (like this one for example), they've all been recorded from my home, and you can see my general set up in this article Practicing What I Preach - Home Recording. From that you might be able to see what some simple treatments might do to help treat for reflections, but this obviously does nothing to prevent sound from leaking in and out.