Look, your not going to use a 120 v to 12v transformer to get 24 volts. Go purchase a transformer to do what you want it to do and quit messing around. You would need a step up transformer 12v to 24v to get to the correct voltage, so nip the middleman out and purchase a 120v to 24v transformer. While your at it go ahead and purchase a 120vac to 24vdc transformer. If you want to mess around with electronics/electricity go ahead and have fun, but your going to have to pay for playing and it would just be cheaper to buy what your looking for in the first place. I know that you can pick up a lot of used laptop electric cords that transform down to about 19vdc, that ought to be close enough for your purposes. You can also buy some full electronic test kits for under a couple hundred bucks to mess around with the basics of electricity.
Good luck, God Bless.