Since large budget projects are mostly performed in a computer just like low budget ones, as soon as you get beyond the basic recording phase of the project (where number of inputs is an important factor) you can achieve similar results.  Granted your vocal booth closet may not look as pretty as the expensive vocal booth in the expensive studio, but you can get comparable sounds from either.  The big differences will come from mic selection, A/D converter quality, and mic pre-amp quality (which in low budget land means nothing as you are stuck with the mic pre-amp built into your A/D converter).

Always record as purely as possible, which means try not to go through too much stuff you are not using.  A good example of this is the guitar player who is plugged through 12 boxes with only 1 turned on.  If you remove all of the boxes not being used the sound will be much better.  Likewise if your vocal microphone is going through several devices before the being recorded you will be reducing the quality of your vocal sound.  The BEST sound is a straight wire.  Try to put as little in the way of that wire as possible.

Also, try not to overly process any sound you are recording unless you are firmly committed to that sound being in the final song.  Especially in the case of processing for the purpose of controlling sound (compressors and gates) effects should be used sparingly when recording.  You can always add the effects later, or even when playing back the recording.  You can never take away an effect once a sound has passed through it.

Be careful when adding EQ.  Do not use excessive EQ in the midrange.  Many natural instruments (especially voices) will change in the tone while getting louder and softer.  The midrange EQ you may add or remove might sound good at one instrument volume but bad at another.  You can always do more EQing later but you will be stuck with whatever you record.