There are four bits of advice that I think have most improved the quality of my mixes, I'll quickly go over the first three and then focus on the fourth which is the topic of this post.
(#1) Use the best monitors you can afford (if you are going to splurge on any music gear, it should be your monitors.) The importance of a good set of monitors is difficult to overstate.
If you can't accurately hear how what you're doing is effecting the sound of your mix then it's very unlikely you're going to be able to create a decent mix that translates to various playback systems.
Don't use headphones to mix. Yes, many people listen back to music on headphones now but what I've found is that if you can make your mix sound good on monitors than it's going to sound good on headphones too but the reverse is definitely not the case.
(#2) Check your mixes in mono too. Even if you don't think your music will ever be played back in mono (and you may be surprised at the scenarios where it may be) I still think it's a good idea to check your mixes in mono.
Using one Avantone Mixcube as my mono playback system (and clicking on the mono playback button on Reaper) has been an ear opening experience. What I tend to do is get my mix sounding pretty good on my stereo monitors and then I switch to mono playback and I usually notice some issues that are not as obvious in stereo.
I think of it as a "microscope" for my mixes. It allows me to hear all of the problems much easier which is awesome because it also lets me fix them much easier! After I get my mix sounding good in mono on this "crappy" speaker it then usually sounds really good when I switch it back to my stereo monitors.
(#3) Acoustic Design. Those of us with home studios usually have less than ideal spaces to mix in but there are some things we can do to improve that space for mixing (and recording.) This is too big of a subject to get into in detail here but I would recommend buying this book and learning a bit about it. It's important.
One easy thing you can do to lessen the impact of a subpar listening environment is to monitor at lower levels (this is also good to decrease the effect of ear fatigue.)
(#4) Cut it out. This brings me to my main topic for this post; the importance of muting unneeded tracks and what may seem like drastic use of EQ cuts to those who haven't thought of using EQ in this way before.
(a) Just because it was recorded doesn't mean it needs to be in the final mix. I almost always find at least one part that can be cut out entirely. If the part isn't essential to the song; cut it out. Hit the mute button. Often times just muting one part makes all of the other parts sound better.
This goes back a bit to my point about mono playback. When you playback in mono it's much more difficult to get everything to sit together. I do a lot of my muting & EQ cuts when I'm playing back in mono. Then when I go back to stereo I'm always amazed how much more clear and "professional" my mix sounds with the mutes & EQ cuts I've done in mono.
This brings me to another point which could really have its own header because of how important it is: Really listen closely to recordings that you love (or the songs you are trying to emulate) and notice how much (or how little) is actually going on. In most cases there's less going on at one time then you probably thought there was.
(b) Massive EQ cuts to the lows & the highs. I put a cut on the bass on almost every track (even tracks that don't sound like they have any bass often do have some rumble that's sneaking into your mix and muddying it up.)
The easy way to do this is to cut the EQ up to the point where you can hear the effect and then lay back off it a bit until you can't hear it anymore. I do this on both the highs and the lows. Generally I hear the cuts on the highs more quickly than I do on the lows.
You have to be careful here not to overdo it to the point where you make the parts sound overly manicured although you should also keep in mind that it doesn't really matter what an instrument sounds like when it's played back solo, it only matters how it sounds in the context of the song with everything playing back.
This is a very common mistake! People will spend hours getting all of their tracks to sound perfect when played back on their own but then they are dismayed when they play them all back at once and it sounds terrible. Think about what the word "mix" means and it becomes obvious why this is the wrong way to go about it. To repeat: It doesn't matter what a track sounds like on its own.
The hardest part of this for me is finding a space for your bass and your kick drum. I don't think there's an easy to share formula for this, you just have to try to give each one its own space (and you have to decide which one of them is going to be the lowest part of the song, because they can't both be at the same time.)
The end result of all of this cutting of lows & highs is not a song without any low end or high end but a song with much more clear and powerful low end and high end.
It may seem like having all kinds of stuff going on in the low end at the same time would be the way to have a really strong bass sound, but it's actually the opposite. If you have a strong bass sound you want that sound to have that area to itself rather than letting it get muddied up by all the other crap in your mix.