I’ve heard that it’s actually more important to get a good recording than it is to mix, because mixing can only go so far? Would you agree with that?
Yes, it’s “links in the chain” and it all starts with good recordings.
“Mixing” is “blending” so if you have a separate vocal track, guitar track, drum track, etc., you mix them together. If you are recording a live performance the mixing is happening acoustically so you can just set-up a microphone (two for stereo) and record. (Pro live recording is typically multi-tracked with multiple microphones and mixed later.)
But, with most modern-professional recordings there are effects & editing done at the same time as mixing so that’s done by the mixing engineer and considered part of “mixing”.
If my recording is already ok, then would I not need to actually bother compressing and normalizing it?
You should only compress if it makes it sound better. It is one of the most common effects but mostly it’s just used to make the recording louder without clipping/distortion. It’s not “necessary” and your recording may sound better without it. Live music isn’t compressed and that’s one reason that it usually sounds better than recorded music.
Limiting is a kind of (fast) compression and it’s easier to experiment with because there are fewer settings to mess-up. So, maybe try that first (with make-up gain or normalize after limiting to bring-up the loudness).
If you’re going to upload your recordings to YouTube, they try to standardize the volume so if you over-do the compression/limiting to make your recording louder they may adjust it down anyway. (Since they don’t apply dynamic compression they may not be able to bring your recording up to their standard volume if it’s too quiet.)
Normalization is usually a good idea. It’s just a volume adjustment so it won’t hurt sound quality.
And, if you apply effects that push your peaks over 0dB, normalization can bring your levels safely down so you won’t get clipping. Normalization is a good last step.
Speaking of levels & normalization - Mixing is done by summation so if you are mixing two or more tracks that approach 0dB, the mix will exceed 0dB and you can get clipping. One solution is to export as 32-bit floating-point WAV which can go over 0dB without clipping. Then, re-import the WAV file, normalize to bring down the levels, then export to your desired format. (You shouldn’t distribute or upload a floating-point file that goes over 0dB because your listeners can get clipping/distortion when they play it.)