Tom pretty much outlined how to test everything but the TIG gas solenoid/HF and you can fire the TIG torch without trying to weld and hear the HF buzz and the gas solenoid click. That would finish it all out as best as you can with no argon.

Fire up the plasma per Tom's post.

Then get some 6013 or 7018 electrodes $5-$10 (you can even get some at Harbor Freight on a weekend). Run a bead in DC and AC (make you you put the clamp on for all modes and in the + terminal).

At this point you know the inverter is fine in plasma and weld mode (TIG and stick).

Now fire the TIG torch (in the air do not try to weld) while in TOG mode and listen for the solenoid to click and a BUZZ sound from inside the machine (front left side).

That's about it. Buy some argon and use the TIG, no mix gas, you can add helium later when needed, but it is not needed for you to start and learn.

Make sure you have the proper safety gear and fire equipment before you start, also I would recommend one buddy to watch with a fire extinguisher near by. No gas cans near by or beer until you're done.