Actually, there are two tools that you need that have not yet been mentioned. When you inflate the new tire, you want to constrain the tire around its tread, so that, when you pump it full of air, it doesn’t over-expand, leading to inconsistent tire diameters and possibly damaging the tire. There are a couple of different versions of this that you could use. The first is a hard tire band. Two of this type can be found on this page:
There’s also the soft trap. The best strap that was ever made was constructed by Exotic Toy Builder, who also made the Tire Whiz mounting tool. If you ever come across something made by ETB, buy it, as he no longer makes kart tools and his stuff was always top-notch. Since you can’t go back in time, here’s another example of the soft strap:
Some people just use a tie down strap. If you go this route, make sure you put a towel between the tire and the ratchet, so that it doesn’t dig into your tire.
Another tool you’ll need is some sort of waterless lube, to lubricate the inside of the wheel so that the tire will bead up nicely. There are dedicated lubes for this, but you can also get away with WD40, as it contains no water, and also is designed to displace water.