A little bit of processing goes a long way, don't over do it. Use eq to cut and not so much to boost. Initial recording counts for the most, get a clear recording at good level or else no amount of trickery will correct it. Look for eq presets and subtle reverb I rarely use anything else.
Curtains are good cheap deadening material. Or covering things with duvet/towels/sheets works as a quick fix. Remember sound waves bounce so a nice shield is a necessary tool to record in one direction (pillows work) and pop shield helps.
Yes, initial recording makes a huge difference.. This became much clearer for me last week when we recorded an acoustic guitar. No matter how much EQ's I did or noise gating, the moment I put reverb it all came back.. So make sure you record as cleanly as possible, then you won't have a hangover from doing repairs