some of the componenets will not work as they need higher voltages
The meter won't show everything. An oscilloscope might.Try some decoupling capacitors.http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html
If the 328 works with a separate 5V on it, but not when running off the 7805,then it sounds like some of the other components in the circuit may be crowbarringthe 7805.
Just a thought about voltages in your circuit.Your transformer will give a peak voltage of 1.414 x 24 volts ( 33.936 ) at the rated output. If you are drawing less current, which seems likely, the peak voltage will be higher. The spec for a 7805 is maximum input 35 volts, your rectifier will drop a few volts but it might still output more than 35 volts so it might keep shutting down.
or add your own winding for a low voltae tap