A few approaches:
1. Write the music so everything meshes or compliments each element. . . I know it sounds obnoxious to suggest this but think about jazz or classical music. They effectively mix themselves with the way the music is written: So much so, that people who record jazz or classical very rarely do much to Affect the mix of the recording. They pretty much just try to accurately capture the performance in the best fidelity/micing they can. Even if you're writing rock, or rap, or country, or whatever other kind of music, if you think about the songwriting stage as part of the mix, it'll make your life A LOT easier down the road. If you're not writing the songs, you can still sometimes make suggestions to the artists in the tracking stage. Planning ahead is going to be your best friend no matter what you're doing. The mix doesn't start with mixdown. . . The mix happens from pre-production all the way till you ship off the final product.
2. Cut away un-needed frequencies. For example, cutting out a lot of the really low frequencies in a vocal take (if it doesn't change the timbre of the voice will help a lot. When soloed you might not notice anything, but it means less tracks competition for the same frequencies. A good rule of thumb is to avoid EQing with something soloed. EQ it as it sits in the mix.
Keep in mind, when you're mixing down to 2 tracks, there are only 2 speakers that can only move in one direction at a time. If you have more than one track trying to fight for the same frequencies, then it can turn to mud. If you're going for a sound where you can hear everything clearly, you may need to treat it a bit like a jigsaw puzzle. . . reduce the amount of overlapping frequencies wherever possible.
3. Play with phase. You'll recognize it as a button with a circle and a line through it, kind of like a no-smoking sign. That's a button that will flip the polarity of the audio single. It won't sound any different soloed, but if you have frequencies clashing and canceling each other out, then you can sometimes find your solution by flipping the phase of a track/or two. Sometimes it works, sometimes it doesn't, but it's always worth trying as it only takes 1 second to find out.
4. Don't try to make everything clear and/or loud. I've heard it said by a number of people over the years that if everything is emphasized, then actually nothing is emphasized. Some parts may not need to be front and center, even if you really like them. Are they essential? Are they just as effective barely audible? If everything's competing to be front and center, then how can anything really stick out? Dynamics in music can also apply to how you mix. Sometimes knowing what to take out can help just as much as knowing what to put in. Guys like Rick Ruben get paid big bucks to do just that (intelligently take elements out of a song to make it better). For example, when he got the gig to produce Johnny Cash's later material, his approach was to literally strip everything away other than the raw voice and raw guitar. He used minimal effects, and didn't touch any delays or reverbs. Johnny Cash had never recorded anything that way before, but it brought out elements that his other music didn't, just from approaching it in a different way. There's absolutely nothing wrong with "less is more".
