It really depends on what you are writing for and the style of music you compose as to what audio software you will need.
The general answer is you will need some kind of sequencer (software to record and edit midi and live parts).
http://en.wikipedia.org/wiki/List_of_MIDI_editors_and_sequencersReaper is one that a lot of people use as it's quite affordable and has quite a lot of the features of highend professional digital audio workstations (DAWs).
http://www.reaper.fm/purchase.php ($60 for non commercial - self use).
Use whatever helps you write your music is what I say. I've used not-so-known software to write music for a lot games such as Renoise.
Software instruments (VSTi) are what many professionals use. They can be either full synthesizers or sample based libraries. Again, it really depends on what style and sound you're going for.
A good general swiss-army-knife vst I like to use is HalionSonic 2 which can get you started quickly.
For a good vast library of musical instruments and percussion there are two major ones people gravitate towards:
Native Instruments :
http://www.native-instruments.com/en/products/komplete/East West :
http://www.soundsonline.com/CCC3-GOLDIf you are writing electronic music:There are actually a lot of free synths and effects out there. KVM-VST website has a great library of knowledge and reviews:
http://www.kvraudio.com/q.php?search=1&q=freeModular and Step Sequencers Renoise (
www.renoise.com) - one I've used for a lot of commercial projects but with my own libraries and samples and offers a fully functional non-time-restricted version for personal use. The only restriction is you can't render out your music to disk.
FruityLoops (
http://www.image-line.com/flstudio/)
Reason (
https://www.propellerheads.se/reason)
Hardware, you'll want:
A Decent pair of headphones that you can mix with.
BeyerDynamic DT770s :
http://www.sweetwater.com/store/detail/DT770pro have a great review, if I was to buy a new set of headphones, I'd give these a try.
Senheiser HD595 (I use these myself) or HD598 (same spec)
http://www.amazon.com/Sennheiser-HD-598-Over-Ear-Headphones/dp/B0042A8CW2Much later or if you can afford it : studio monitors (speakers). However, make sure your room has some acoustic treatment first. There's no point blowing a lot of money on some good studio monitors if your room is untreated. It's like buying an expensive monitor with a red room light and a blue one in the corner, it'll throw the color calibration off and your screen will look wierd. I actually wrote most of my first game compositions entirely in headphones.
For room treatment, an Auralex Room Kit will get you started.
http://www.sweetwater.com/store/detail/RoomDST36COne of the great things these days is there is sooooooo much used gear that you can get started at a fraction of the cost of say 10 years ago.
However this being said, you nailed it on the head - write great music, that is high in production quality. Rather than spend 4 hrs playing a game or TV or browsing the web, spend 1hr doing something like that, and 3 hours writing music. The more you write now, the more investment into your future you'll have. I started composing in highschool and didn't stop and spent most of my spare time writing music.
If you already play an instrument practice
improvisation, take a song you like and try to write in a new melody, or slighly change the notes as you go. Improvisation is basically micro-composition with an existing song structure. Take Twinkle Twinkle Little Star for example, using the same chord structure and melodic content, try coming up with a new melody that is slightly similar to the original or completely new.
If you already play an instrument, practice a LOT. The more familiar you become with your instrument, the more you will just become acustomed to it which will make improvisation and composition easier. While computers have made it easier to write music, there is no substitution for being able to bang out a melody quickly then revise your idea in real time on an instrument. If you don't play an instrument already, start learning one. Music makes more sense when you play it.
Think of it like Walking. If you start to learn to walk, walking sideways and backwards, and walking pigeon toed, and waddling isn't easy. But if you walk a lot every day, and get used to it, you can easily do any of those things with your eyes closed now. That is how music works too if I can put it simply.
Answering your audio engineering question:
So whilst you don't
need audio engineering to be a composer it certainly be of huge benefit in understanding how to mix and produce your music. A more polished product is going to help get you work.
However that being said, I work with high end composers who haven't taken audio engineering courses and have just figured out how to mix and produce their work through self learning and practice / experimentation and hard work.
I use audio engineering skills on a daily basis as I now primarily work as a sound designer and have to do things like mix trailers, and produce voice over, creating and mixing games. It's just another skill to have which will help you in your career path.
I actually took a course when I first started getting work in games to better my knowledge and have a formal qualification under my belt.