For the quickest acceleration and sharpest handling you want the lowest possible weight at the rim, ie. farthest from the hub.

A heavy rim, tire or bead, takes more of your energy to accelerate, like a fly-wheel in your car: a light flywheel accelerates quicker, but a heavy one has more momentum once it's rolling.

Sew-ups, or "Glue-on" tires, are lighter than clinchers, but are generally more expensive (and messier) and are smoother riding and generally faster.

A kevlar bead (not belt), as opposed to a metal bead, on a clincher is a compromise - it is lighter and easier to install, but usually more expensive.

Kevlar "belt" tires are supposedly tougher more flat resistant tires, better for training than racing. They may be considerably heavier than sew-ups, and therefore slower to accelerate, taking more energy from the rider - important for short course or sprint performance, but only if you are already a contender. For anybody less than an expert rider, who has a chance of placing, the seconds you save with a "Formula One" tire are outweighed by the potential loss of minutes, fixing a flat.

Psi (max) = pounds per square inch or tire pressure, and is determined by the manufacturer. The higher the pressure, the faster the ride, but.. out on the road it can be difficult to get really high pressure into the tire after a flat. CO2 cartridges are a nuisance and expensive if you flat a lot, and nobody carries a foot pump out on the road.

We all want the fastest, smoothest tires. 2 factors determine the choice for riders: cost and convenience.

Size is about 2 things: 700 refers to your rim size
18, 20, 25, as in 700x18 or 700x20 refers to the profile of the tire, ie. its height above the rim when inflated.
Lighter riders can ride on 700x18 tires. Clydes like me need 700x25s on the back, or else we get a lot of pinch flats. A lower profile tire is usually faster and more energy saving, which means a hard low profile tire is also a slightly harsher ride due to vibration.
Bikes with 700 on the back, and 650 on the front, called "funny bikes" were popular for a bit, but proved a headache because you must carry two, different sized, spares.

There are tread pattern differences too, but this is largely irrelevant to anyone less than expert racers. One exception to this is the weather you usually ride in. tri-bro lives in Oregon, he gets a lot of rain I bet. Slick treads are fine on dry roads, but will wipe out on wet roads.

Once you have decided on the tire system there is nothing to choose in quality between the top manufacturers your shop carries.

regards,
PoC

Guys, thanks for the information tips. PoC are you using new tires for your IM? How often you change tires? Does the rule of no new tennis shoes for a marathon applies here. No new tires for IM race?

I just got my first bike (Trek 1200C) for Christmas and I did buy a small hand pump that mounts under one of the water bottles....I read that "nobody carries a foot pump on the road" from PrinceofClydes...I don't want to look like a dork, so should I take it off my bike?

I have one of those mini-pumps too. It mounts on a clip beside my downtube water bottle bracket.

I hope I was not being unclear. I use a handpump on the road. For training I might even carry the mini-pump, but the one I carry in races is a full-length handpump that fits under my top tube.

The foot pump is that thing you stand on between your feet and use one, or even two hands to lift and press down on. They work great and I see people using them to top up their tires pre-race but are way too bulky and heavy to strap on the bike. That's why "nobody carries a footpump on the road." :)

Try stuff out on training rides and find out what you are most comfortable with.
And never worry about looking like a dork by having equipment that works. Jimmy Riccitello, a top pro, was standing on the side of the road during the St Croix Half IM, begging passing riders for a spare tire and pump because he neglected to pack his own! He had been leading the race when he flatted. Who looked like a dork that day?

PoC

RE: Brian B post for gear,
For a half IronMan and longer, as well as weekend long rides, I like to carry 2 small tire levers, 2 tubes
saturated in baby powder in a small plastic baggy, and 2 CO2 cylinders and a CO2 trigger actuator..
You would be amazed how little space this takes up, when I place these items into a small, clear, thick plastic bag, and roll all these items up tightly (like a sleeping bag) into a small roll. It is then secured
with velcro straps onto the back of my XLab saddlewing (water bottle holder). After flatting 10 miles into a half-ironman, I didn't have to hold my breath and pray that I would not flat again in the next 46 miles, which could be a race-ender for the "1 tube-1 CO2 cartridge" carriers. Hope this helps. PS, I highly recommend the XLab saddlewing. I love mine, especially with the added "x-hook" that allows you to hang your bike by the rear-end of the seat, so you can pull it off the rack faster.