Argon. Unless you’re specifically doing something really abnormal there’s a reason it’s the most common shielding gas. Compared to the rest of the process the shielding gas shouldn’t be much of the total cost. If you want to minimize cost don’t rent your cylinder. You can usually find a cheap one used and just exchange it on your first refill if it’s within test date.

Nitrogen or CO₂. But you will not be happy with it, there is a reason more expensive mixtures are preferred. Also, the required protective gear against the NOx and added postprocessing of your workpiece more than make up for the gas price...

The cheapest way overall is actually the commonly used Argon. Its just not cheapest if you just stare at the price of the gas alone.

A lot of times the supplier matters as much (more than?) the gas. Ask around in your area.

Avoid bottle rental at all costs if you can... buy your own or bottle swap are the best.

Agree with sibling poster; just get argon for mild, stainless or alu. It is absolutely the standard for the tig process.

Most times gas is not an appreciable cost, usually time or materials but I guess it could be a factor if you're welding up a lot of scrap and count your time as free.

I haven't added it up but I think I spend more on grinder discs than gas :p