I learned to do it years ago at an 'old school' Amoco station in Miami. The local racers would bring their racing tires and magnesium wheels in for us to do by hand. (Think 10" Minilites for a Mini)
Not too hard once you learn the trick of it.
You need some real tire irons. Not the bent rod, chisel on one end, lugnut wrench on the other that come in the trunk of a car. Real tire irons are steel rods about 1/2", tapered and flattened on one end into broad, flat spatulas, with all the edges smooth. But you can alter a couple of big flat balde screwdrivers by grinding and smoothing all the edges. There should be no sharp points anywhere to catch or cut a tire, or mark a rim. You also need some tire lube. (Yes, KY will work) You can buy it, or make a weak soapy solution of liquid detergent.
Disclaimer: This is the way I've done it for 40 years. YMMV. Please take all appropriate safety measures, and work safely.
Every normal wheel is designed with a depression or valley around the center of the rim. This allows the tires to mounted, by providing a place for the bead that allows the opposite side to slide over the edge of the rim.
Start by placing the rim on a flat clean work surface, on the side that places the valley closest to the top of the rim. Slap some tire lube onto the tire beads. Lay the tire onto the rim, and push one side of the bead down and over the edge of the rim, and into the valley.Push the bead down and around as far as it will go. Hold the tire down onto the rim with your knees, and slip a tire iron between the rim and bead on each side, about 1" in from the point where the bead is past the rim. Hold one iron in place with a knee (you have 3 knees, right?), and then lever the bead up and over with the other iron. You shouldn't have to use a lot of force. If you sense that the tire is being cut or gouged, stop. The bead may have slipped out of the valey, or you may need to take smaller bites. Use the 4th knee to hold what you just gained, and move about 1" in, slide in the iron, lever it up and over, ...Continue until you get to the other iron, and lever it up and over.
Repeat with the second bead. You may have to crush the tire down to get the bead into the valley, that's OK, it's made of rubber.
Wipe off the excess tire lube and inspect the beads for damage and debris that may have stuck.
Now, to seat the beads and inflate the tire. CAUTION Compressed air is like a spring, it can store a tremendous amount of energy, and release it suddenly and unexpectedly. Never put your fingers (or any other appendage) into the bead area when inflating. As much as possible, keep your distance (use an airchuck clip), and wear eye protection. When a lower bead suddenly seats and pops into place, the tire and rim can leap into the air and hurt someone.
Try to seat the beads and inflate the tire. If that works, cool. Otherwise, you have to mess with it some. You want the greatest inrush of air you can get. Remove the valve stem core. Slather some tire lube onto the beads. Use the shortest/fattest hose you can get. Start with the highest pressure you can. If that doesn't work, wrap a ratchet tie-down strap around the center of the tread, and cinch it down, that will help to hold the beads apart. It can help to raise the rim, to allow the tire sidewall to fully extend, another rim of equal or smaller diameter helps here. Once the beads do seat, stop, remove any devices, re-install the valvestem core, and then inflate to the correct pressure.
I hope this helps, video would probably be a big help ( 1 pix = 1K words ).
Now, breaking the beads loose to remove a tire can be really challenging. But that's another thread.
Carter shore