Totally agree with cmbroom about grad school (or postgraduate degrees as we call them in the UK) being about "pulling from within". If you start doing this as an undergrad, it's easier to get into the habit.
I did my MA when I was 21 at one of the older, more recognized universities in the UK. Hard to say which schools have the best courses - this really depends on your research area. They vary from place to place. Usually if you pick a university with a 'name' it is naturally assumed that your degree is decent or better than the next person’s.
In the UK we have league tables that rate universities and departments. Picking one that comes in the top few of the league tables, if you can secure a place, will ensure that your degree is considered some what more prestigious than one from a less well established uni regardless of course content. This is what I did for mine.
If you're doing a Ph.D. (I began my doctorate directly after my MA at the same university and then gave up my Ph.D. during the very final year, just before my thesis was due because of serious health problems – it was a shame and a big disappointment for me) then it helps to pick a place where you have a supervisor who not only has the knowledge to guide your research but who is also well-established within the field because this will give you advantages with regards to contacts, getting your work recognized etc.. This is what I chose to do and I can honestly say it helped my research and the quality of my work. Picking a supervisor who you like and trust is important too because you’ll probably be working closely with them and taking a lot of constructive criticism.
At doctorate level networking is important ( particularly if you’re planning a career in academia) so is getting used to teaching and basically getting your work out there, getting involved in seminars and conferences as often as you can and doing anything to stop the onset of ennui during your final year.
Postgraduate degrees are fun, more so than undergrad degrees because you have more control over your work and can pick your own direction. It also involves working alone a lot, so if you enjoy this then it’s ideal, providing you don’t become a hermit and avoid things like networking and so on.
Hope this helps.
pooh_bah