Replacing tyres is quick, easy, and can massively improve the ride of your bike.
Remove the wheel(s) of the bike you intend to operate on. If it's a normal bike, this will done with just a 15mm spanner.
Fully deflate the tyre, and then try to pull one side of it off the rim. If this is a particularly tight fit, you might need some sort of lever to wedge between the tyre and the rim to pop it over (spoon handles work a treat!). If you use a lever, take care not to pinch the inner tube between it and the tyre. Doing so can cause a very hard-to-fix puncture.
With one bead (the wire inside the tyre that holds it on the rim) removed, push the inner tube valve out of it's hole, and remove it from the wheel. Then remove the other bead of the tyre, and it's off!
Some tyres are designed to work in one direction better than the other for improved grip/speed. Look at the sidewalls of your new tyre, and identify which way around it needs to be (if any). Place one bead on the rim.
Whilst one side of the tyre is fully on the rim, and one fully off, re-insert the inner tube valve through the valve hole. If you have a dust-cap, put it on the valve once it's through the rim to stop it popping back inside the tyre before you get a chance to inflate it.
Stuff the rest of the inner tube back inside the tyre, and pop the other beading back into the rim.
The last section of the beading is often quite tight with new tyres, so you might need the levers again. (Be sure not to puncture the inner tube, again!)
Inflate the tyre, put the wheel back in the frame/forks and roll on!