What do I connect the ground and power symbols in this schematic to?
There are 5 grounds and 3 power source (12V).


Generally you connect all grounds together.

You also can connect the 12 V lines together but in this case you have a very high powered electro-magnet. The current draw and switching noise from this may cause interference in the rest of your circuit.


  I bought a 12V and 1A wall adapter. It has a plus and minus end. Can I connect the other 2 12V sources in parallel. Will the reduction in current affect this device.


There won't be a reduction in current by paralleling devices. There will be an increase - and a large one.

From Ohm's law you can see that the current is given by \$ I = \frac {V}{R} = \frac {12}{2} = 6 \; A \$.

Your 1 A power supply hasn't a hope of driving this.