Should I solder multiple wires together, or is there a better way?
You could use a piece of stripboard to solder the wires to, and some PCB screw terminal connectors for a neat, disconnectable connection to the power supply.

You might need to drill slightly larger holes for the PCB screw terminal connectors.

If you solder all the wires together in one lump, it will be difficult to remove one if a replacement is needed.