AAC and its successor HE-AAC provide very good audio quality, even at low bitrates. In that respect, MP3 isn't that bad either, given that you can use Variable Bitrate Encoding (VBR) at bitrates from 198 to 320 kBit/s. Variable bitrate ensures that bits are allocated to those parts that need it, and it conserves bits for passages that are easy to encode. Don't use Constant Bit Rate encoding unless you are in a streaming scenario – it'll hurt quality.
Note that you should check which encoder you are using. FAAC and LAME are typically considered good (and free) encoders, but that doesn't stop you from trying others, as @Lèse majesté pointed out. For example the Fraunhofer MP3 encoder, which is commercial.
This is a subjective thing – and if you're passionate about music, just trust your ears and try for yourself.
If you're unsure, this figure compares some recent audio codecs at 48 kBit/s – thus, a really low bitrate. It's taken from an EBU technical paper.
However, what you never want to do is reencode audio files from another compressed format. Say you already have tons of 128 kBit/s MP3s, they won't sound better in AAC. They'll even sound worse because you're throwing away information from a file that already lost most of its quality.
So, if you transcode, then only do it from WAV files, CDs, or other lossless media such as FLAC files.
Trust me, I was so stupid as to rip some CDs to Windows Media Audio back in the days. I later had to convert those to MP3 again, and you could definitely hear that they were really bad in quality after that process.