How to make my music or audio project sound good almost everywhere ?

Well, this really takes a long time to get it right.
But the main formula for people who don't make a living out of this, is, first to have a decent reference. This can be a good headphone, or a decent pair of speakers.
Then, search for a music, presentation, or something that you like the sound, and try to copy that.
Listen carefully to the volume differences of the different sources (could be instruments in a song, or speech and background music in a presentation), and try to apply that in your project.

You can also play with EQ. Just open an graphic equalizer (they are present in my audio products) and start playing with the different bands. Note, that sometimes, you can add high frecuencies (or treble in other words), to a track. After a while, your ears get used to that, so you add even more treble. And it sounds good at that moment.
When you come back after a while, with your ears relaxed, you will note that the track has too much treble.
So again, always use an audio reference to compare your project while working.

Finally, once you finish you project. Try playing it everywhere, in your car, your stereo, tv, whatever. That helps to hear mismatch in levels, or EQ problems.

Recommended headphones/studio monitors

It's good to get a pair of professional headphones, so you can hear your work as is meant to be.
I recommend the Sennheiser HD 280 Pro for studio work.

If you are really serious about the mixing, you can get studio monitors. This is much better than mixing with headphones.

Just get the small ones at first, usually 5 inch speakers.
I use the Yamaha HS-50. I also got very good coments about the KRK.