Here's a fantastic text on the topic:
http://www.clarity.net/adam/tire-changing.html
This is how I've done it for years -- dirtbikes, vintage bikes, modern sportbikes, etc.
It's a skill you have to learn -- the first few sets are very tough. With a little experience, you develop the skill, so try to find someone with more experience to help show you the ropes the first few times.
Honestly, I've used a No-Mar tire changer, and I wasn't the least bit impressed. By the time you finish futzing with the stupid No-Mar wheel clamps, I'm done using tire irons.
Modern radials are actually easier than narrower old bias ply tires. Wide tires do trip a lot of people up because it's more difficult to keep the opposite side of the bead in the drop center. Once you get the hang of that, they're like buttuh.
Two hints that will save a lot of heartbreak -- lots of people try to take stupid shortcuts for some reason on the following two items:
1) Spend the lousy $15 and buy a damn jug of tire mounting lube (Napa RuGlyde is easy to find). Hillbilly substitutes don't work as well and could kill you.
2) Take the time to make a stable, safe, way to SLOWLY and CONTROLLABLY apply force to break beads. Don't use a car, truck, explosives, kinetic force of any kind, hammers, etc. I use a very simple arrangement of two 2X4 boards attached to a leg of my bench. Breaking the bead doesn't take a lot of force -- I can do it with three small tire irons. There's also a Motion Pro tool called the BeadPro bead breaker that works on the same principle.
Also, replace those ancient tire valves.
Anyone wants lessons in Indy, PM me!
Here's my high-tech balancing setup, made with salvaged skate bearings and scrap -- you can balance to 1/8 ounce easily with this: