Along with using a good heavy "tire fitters"rubber mallet, tire fitting lube that I bought from my local tire joint, and a good strong pair of work boots with a sharp heel, the handiest thing that I have for fitting tires at home is this;
I made this wooden frame sometime last century. It allows the wheel to sit flat and not rock around while working the tire on/off the rim. The next most handy thing is a set of knee pads, as the knees are knackered and don't take kindly to kneeling on the ground for long periods.
I haven't had to change any tires during competition for many years, I used to be able to do both wheels on my 1978 390 AMX in under 6 minutes in "Parc Ferma conditions" when it was in 4 Day trim, but it has turned rims. Best time ever for a bike with std rims and rim locks was both wheels on a Yamaha TT250 in under 5 minutes during the Dunlop Tire Challenge at the Royal Adelaide Show back in 1986. I was working for a Honda dealer at the time, but the local Yamaha dealer let me borrow a TT250 for a day so I could practice on the bike to be used in the competition and work out a system for getting the wheels out and back in quickly.