Yes you can build such a thing. I would first look for a big fat wallwart at a thrift shop. A wallwart can do 1 amp if they are of the somewhat bigger variety. The 7812 can do 1 amp, but you'll have to heatsink it pretty well, more if your input voltage is much over 18+ volts. 3-terminal regulators are magnificent things to be sure; But when you use them near the top end of their current ratings, it is somewhat of an art getting enough input volts (probably needs dead minimum 15) so that they do not drop out without getting TOO many input volts, because the input volts need to be dumped through the device at the current you wish. If you need 1 real honest-to-goodness amp and supply the 7812 with 22 volts, that's 10 watts the 7812 has to dump. That implies a serious heatsink.
The real beauty of 3-terminal regulators is when you need very low power and you can just slam them into a circuit with (only) their own tab as the only heatsink. Easy as pie. Once you start to get close to their ratings, they need heatsinking.
I would also not rule out an open-frame Power One or Condor (or any of a dozen other brands) power supply from ebay. You find these for sale for $5-10-15 all the time. They are pretty bulletproof. The problem is that the cost of all the parts you need plus the assembly effort for one you build from scratch is generally a lot more than just buying one of these things. Seriously, can you go buy a suitable transformer for $10? Never mind the caps, the regulator, etc etc. Search ebay for "Power One 12 volt supply" or "condor 12 volt supply". There are loads of them.
http://www.ebay.com/itm/Condor-Power-One-HAA15-0-8-A-DC-Power-Supply-/180983770218?pt=LH_DefaultDomain_0&hash=item2a23792c6a Finally, If you decide to build the thing yourself, suggest you get a 7812 in a TO-3 case, the type that looks like a 2N3055 transistor, versus a TO-220-cased one. Generally easier to heatsink.