I recently purchased a cylinder of MIG shield gas (argon/CO2). Can I use my existing flux core wire and the MIG gas together or do I need to buy solid wire?
Maybe, but best not use the gas. The flux core has deoxidizers in it like aluminum , silicon and manganese ( not really a deoxidizer). They are expected to react with oxygen from the air and be consumed. If the oxygen is kept away by the gas some of those elements will dissolve into the molten metal . This will add alloys that may affect the properties of the deposit . Many years ago Lincoln did something like this and the resulting welds had high silicon and manganese causing them to be too strong.
You can use both but as tester101 mentioned it is a waste is $. I have put my emergency roll of flux core on by mistake it welded fine with the gas on. Now that you have gas get some solid wire. Solid wire with gas is so much cleaner and no flux slag to chip.