Taking the old headset out is pretty easy - just a long screwdriver and a hammer. make sure you hit the inside edge of the cups at several points, to keep it coming out straight. I've found the proper split-pipe headset removal tool is only really needed if they're very very stubborn, or have a thin walls that a screwdriver can't hit against easily. Taking the crown race off is also usually pretty easy - same sort of method really.
Installing a new headset can be done quite easily. I have a homemade press that's just a length of 8mm threaded rod and a selection of washers and nuts. Just tighten everything carefully and make sure the cups are going in straight. The crown race can be tapped on with a screwdriver, but only in the lip, don't tap the bearing race itself!
I don't know what thread your raleigh will have - probably english, but I know raleighs sometimes used their own thread sizes - there are plenty of experts on here who know more than me and should be able to tell you one way or another!
Stack height is important, but only really limits you if the headset you want to use is taller than the one you have. You can use a shorter headset and add some spacers between the race and locknut, that way you don't have to cut the steerer and have more options should you need to change again in the future. On my Nortelle I'm stuck with the ultegra 6400 headset it came with, as it has a very low stack height, and on top of that is french threaded...
You might find your existing headset just needs a good clean and regrease, maybe with new ball bearings. You don't need to remove the cups or crown race to do it, but you do need to remove the forks.
When adjusting the headset, I always find it's best to tighten down the upper race just a bit too far (maybe a 1/4 turn), then tighten down the locknut before backing off the race, locking it against the locknut. with a bit of practice it's easy to do, leaves the race locked tightly in position but with the headset perfectly adjusted. You only need one headset spanner with this method too