Any reason for wanting to stick with the original headunit? There are aftermarket HUs made specifically for the A3 so they fit perfectly and if you want you can have a 7'' touch screen, GPS, bluetooth etc.
I have ordered the one that's on Xtrons website but you can get them on ebay for £300 new or one seller puts up a couple of refurbs each week and I just picked one up for £200. The refurb only has 45 days warranty but I wouldn't be surprised if it was a new unit tbh.
The way it usually works is the headunit has RCA outputs, this is a 2 socket cable which feeds your amp with the audio signal. The amp has a direct 12v feed to your battery, there will be a fuse under your bonnet to protect you if something went wrong. The amp will be earthed to a point in your boot. There will be one further wire called a remote wire that goes from the headunit to the amp, this tells the amp when to switch on.
The amp will then have twin sets of connections which go to speakers, a positive and negative to each speaker. A 2 channel amp for example will have 4 speaker connections. You might be able to get a 4 channel amp where 2 channels are bridgeable so the power for 2 channels goes into one and this will power your sub whilst the remaining 2 channels power your new front speakers.
Unless you were on a very low budget you would never use your headunit to amp your speakers, you would use a separate amp as described. You could have a sub box in your boot, I haven't looked at stealth boxes yet for the A3 but I built my own stealthbox out of fibreglass for my Focus.
If you have any questions then just send me a message as this is one area where I can be of use, even if I am a little rusty as my last install was over 2 years ago but I have done quite a few in my time for me and my mates.