After some research online, I do not know whether to use a MOSFET, transistor, or relay. I do not have experience with these and are looking for any suggestions.
Which will work the best and provide enough power?
What power supply (voltage and amperage) would I need to power two of my magnetic locks?
What relay would I use to control both these magnetic locks with my Arduino and what electrical specifications (voltage and amperage) do I look for online when I search to buy the relay?
What else would I need to wire this circuit and how would I wire it?
What power supply (voltage and amperage) would I need to power two of my magnetic locks?
Perhaps you should read the description that you cited? (link corrected)
Voltage:DC 12V
Current:380-430mA
Two times that is 12 V at 1 Amp.
TWA2020:
2. What relay would I use to control both these magnetic locks with my Arduino and what electrical specifications (voltage and amperage) do I look for online when I search to buy the relay?
TWA: you find yourself in a state known as "analysis paralysis".
Find the project hub. Find examples of working magnetic blocks. Use what they used. Do what they did.
A kick back diode was mentioned above. The technical term is commutating diode. Bottom line: find a project that uses them. If you don't, you can make it work but you can't make it last