This topic from Long and Short Reviews was interesting. It's been a while since I've been to any museums or galleries, I have to admit. However, there are a few I think are worthy of mention.
I've visited the Bronte Parsonage twice and would definitely recommend it if you're in the area and have an interest in the Brontes. They have a remarkable collection of Bronteana and the parsonage itself is a beautiful building. Similarly, I visited the Anne Frank House when I was in Amsterdam and found it to be a humbling experience - just seeing how enclosed their hiding place was is really quite eye-opening (although please don't go there with my ex, who stood around saying "Can we go now? This is boring.")
As for where I want to visit, I plan to go to the Richard II Museum soon (I'm a Ricardian all the way) and would love to see all of Vermeer's paintings, if not for the fact that they're all over the place.
I'm interested to see what everyone else will mention!