I work as a freelancer in Austria (web design. mobile).
I was already an EU citizen so that made things slightly easier, but the process itself is incredibly complex and slow, mainly due to the fact that there is very little in the way of initial assistance if you don't know where to begin. I had to learn that the hard way, so hopefully this information will help you to get through some of it:
You'll need a German speaker if you can't speak the language yet.
First visit your local WKO (Wirtschaftskammer). They will go through the basics with you and you will need to sign LOTS of forms. You will also need ALL of your identification: passport, proof of address (multiple), a police record either from Vienna or from your country, your birth certificate (required for the police record) and marriage certificate if married.
The WKO will charge you a yearly fee whether or not you use them after the initial set up - there is no way around this. It's around €100 per year.
You also need to pay a yearly business fee and a tourism tax fee (pricey vary on your location).
The first tax bracket starts quite low (around €11k), and the tax is very high (currently 36.5% for even the lowest income earners above the non tax bracket) but apparently this is being reduced from Jan 2016.
Social security is also very high, but unavoidable. The cost fluctuates a little in the first 3-4 years (get's cheaper in the third year then more expensive in the fourth). You will receive the social security forms in the post a few weeks after registering with the WKO (weeks if you're lucky - I had to wait 2 months). Social security in Austria, as with Germany, is vast and complex. Don't try to fill this form in yourself. Take it to a.......
(Find yourself a) Steuerberarter (tax and finance assistant) - this isn't a legal requirement but the process is so complicated it's pretty much impossible to do it all yourself. Warning, they are not cheap but they will handle all of your finances for you: Tax, social security and VAT (they will more than likely insist you set yourself up with VAT as the starting bracket in Austria is much lower than most other EU countries - again you have to pay to register for this).
VAT and social security is paid quarterly. Standard tax is paid yearly from the date you started.
Prepare yourself for a lot of junk mail both in your post and your email. And check with your Steuerberarter or WKO every time something comes through the post that is asking you for money. Not everything is mandatory (although sadly most things are).
In short - don't try and do it yourself, get help, Austria hasn't embraced the digital work of forms yet, and expect to spend a fair whack of money both registering and paying taxes.