Choosing between relays and MOSFET driving multiple loads
You want either MOSFETs or mechanical relays.  Solid state relays usually have too high a voltage drop to be suitable at these currents.  There are solid state relays that use MOSFETs as the switching element.  However, since you don't seem to need the isolation, it's probably easier to use MOSFETs directly.

Switch as many loads separately as you can.  That reduces the maximum current any one FET needs to be able to handle.  You can then choose to switch several loads together if you like.  Minimize the maximum current each switch needs to deal with.

Low side switches will be simpler, if it is OK to keep the + input to all loads permanently connected they don't need to have a common ground when off.