Unfortunately, the Aero Club does not, at the moment, provide flight training. At Canberra, the only organisation actually on the airfield which provides flight traning is Brindabella (who also happen to run an airline). Brindabella is very convenient, has high standards, and, for CPL students, provides a plausible path into airlines jobs. See http://www.brindabellaairlines.com.au/index.php?q=node/237 for details.
Other options around the district are...

• ACTAS  They are actually at Hall, and fly from Canberra.
• Goulburn Flight Training - RA-AUS and GA (45 minutes drive from Canberra)

  CFI Teraya Miller 0434 558 088
  Club President Tim Allen (02) 4821 7798
  Instructor: Austin Cummins (02) 4832 2035 / 0407 321 876

• Moruya Aero Club  RA-AUS and GA - Combine your flying with a weekend at the coast

• Polo Flat, Cooma. RA-AUS only. CFI Michael Apps: 0411439197 / 64526636, Alan Middleton 0407356948, or Paul Griffin 0448604082 (these two are RA-AUS instructors and also members of Canberra Aeroclub)  - probably not your cup of tea if you are completing a normal GA PPL

• Slightly further afield, Camden and Bankstown airports in Sydney offer a number of schools. The drive is 2.5-3 hours from Canberra.