It was odd walking through Westminster Abbey; I got rather angry at the other tourists. The place is actually rather small, and you have to walk in a line through the outer loop. The problem was that there were people chattering away, and fake high-fiving dead kings, etc. Not that tourism should be a dry formal event, but come on; have a little respect. Ok, enough of a rant for me.
Other tourists aside, if you take it upon yourself to be good and quiet, you can have a nice moment in the Abbey. There is a plethora of famous Brits buried there, and history aside, there are lots of cool sarcophagi and the such to look at.
One thing that a fellow hosteler pointed out, is that there is a lovely prayer room, which is on the far side of the poets corner. I went in there, and it was me and one other man, who left after a minute, leaving me alone. It was absolutely dead silent, because it is a huge stone room with an extremely thick door. Again, don't go in here if you're going to be loud, but if you're looking for a nice quiet spot, most of the tourists ignore this room.
When you walk through, be sure to pay attention to the floor, because there will be names of famous people all over the place, and since there is so many, they're not terribly noticeable. (I almost missed Charles Darwin, although I don't think he's actually buried there.)
All in all, it's a lovely place to go. I think they recommend a several pound donation to get in, and there is usually a short line, but it is worth it for the history, and so you can say you've been there.