I'm trying to figure out how to use this forum.  How can I attach a picture?

Thanks!!

Problems -

1. The 5V of TLP621 is reverse blocking voltage of the LED. Use a series resistor minimum 100R / maximum 680R.

2. The Solenoid freewheeling diode is in forward bias. It will conduct current and the TIP120 will be destroyed when the Solenoid get closed. Change the diode direction. Use 1N4007 which is commonly available.

3. I am not sure why you choosed such high power Darlington pair transistor. It will work but it is not for what you have intended to use it. Try proper switching transistor with atleast 1A current rating.

4. If you choose to use TIP120, having 75 ohms base resistor could provide [12V / 75R = 160mA of current if the TLP phototransistor is ignored] a lot currnt to the transistor base. Use atleast 180R to 680R resistor if the current consumption is not a problem. Otherwise, use a Mosfet which will be cheaper and heat dissipation will be lower than high power transistor. If cost of the component doesnot matter, use low RDSon resistance in miliohms rating.

1. & 2. taken care of. Thanks

3. I picked the TIP120 because Adafruit suggested it with the solenoid. (Remember I know Nothing.) Would this be an appropriate switching transistor?  

 4 "If cost of the component does not matter, use low RDSon resistance in miliohms rating."  Could you recommend one of these?  I really don't know what to look for.

1 & 2 points taken care of, thanks.

3. Would this be a proper switching transistor.  

4 Could you recommend a low RDS_on?  I really don't know what to look for.

Is the TLP621 even needed?

You could try STP90NF03L. You would not need a higher size of heatsink.