(Apologies in advance for the fairly long post - it could probably be written a lot shorter, but I'm bad at condensing my thoughts.)
Agree with the importance of counterpoint here. Worth starting with the earlier stuff (renaissance/baroque counterpoint) and expanding forwards, maybe? -- earlier counterpoint is more restrictive tonally and easier to grasp the basics of, and the closer you get to the present day the more directions and possibilities open up, and at the same time the more complex it gets -- 20th Century classical composers like Bartok and Scriabin and, as Alycs said, jazz musicians, often work within tonal systems, but very complicated or open ones.
Probably also worth having a look at the basics of orchestration - the way a piece is arranged, which notes are played on which instruments, which are doubled in different octaves, the different sounds you can get from instruments depending on the range and manner in which they are played... can all have profound effects on the way a piece sounds, especially in more stripped down works. When you're writing for only a few instruments, little effects like vibrato, pitch alterations and volume swells can have a huge impact on how well the piece's emotions are conveyed. Also, have a listen to some chamber ensemble pieces and string quartets, even pieces for a single instrument. If possible, it's good to have a copy of the score to hand as you're listening, so you can have a look at what's going on in the music when you hear something you like and would like to replicate (
IMSLP and the
Gutenberg Sheet Music Project are good for this). If there's music from a video game you like the sound of, see if you can find a midi of it (
VGMusic is the go-to site for this, but if the midi you want isn't there, try Googling) and plug the midi into your DAW, or into sheet music software with midi support (
Musescore is hit and miss when it comes to interpreting midis accurately, but is still probably the best free option out there) and have a look at the way the piece is put together, what exactly it is about the piece that makes it 'emotional', be it chords, the intervals the melody uses, the arrangement... On the subject of suggested reading for orchestration, I don't have any sites to hand (though I'm sure there are plenty), but the book
Orchestral Technique by Gordon Jacob is good.
(Bit of a tangent: Modern/contemporary composers are often very concerned with silence, texture, space and so on, and so are also good for coming up with ideas for interesting ways to use sounds, extended techniques, space and so on to give a piece more character - even if you're writing tonally these techniques can add a lot of character and interest to sparsely orchestrated pieces. In that vein I'd recommend giving a listen to (and if possible, reading the sheet music along with) any of
Toru Takemitsu's chamber work, anything by
Penderecki,
Crumb,
Varèse or really any 20th/21st-century composer who does small ensemble work. Find a piece that does something you like, see if you can get your hands on the sheet music if possible, and try to replicate the sound yourself in the DAW.)
Sorry for the length, and the tangents. Hope this was of some use.
(Edit: partially agree with itroitnyah about the higher-pitched instruments -- that's certainly
one way to do it. Then again, high-pitched instruments can be aggressive and piercing (see the George Crumb I linked), while lower pitched instruments (especially cellos oh god cellos *melts into puddle of goo*) can be so very beautiful (
clicky and
clicky). Depends how you use 'em.)