Note: I wrote this while in Europe with students, for them, and thought others might be interested.
Many people find museums difficult, I think, because they feel compelled to shuffle through miles of works one-by-one, stopping to study each of them and dutifully reading every caption. Maybe this gets instilled in elementary school somehow. Or maybe it's what people feel they ought to do. What a terrible experience! Where's the passion? Where's the fun?
Instead, consider this advice:
Browse quickly and be pulled. You may move rapidly, perhaps not even breaking stride in a room full of masterpieces. Soon enough, something will grab you. There will be a spark, a connection. Stop and spend time with that one. Be submissive; stay in that moment of intimate contact as long as you can. You might make a sketch if the spirit moves you, but draw freely without much thought.
Then when the mood is broken and you disengage, stay with that piece that grabbed you and (only then) turn on your analytical brain. Look at the work again. View it from every angle, and multiple distances. Figure out why it pulled you. What is the work's power? Perhaps it will grab you again!
Then you may start thinking about the usual questions of subject and meaning and technique: what is the artist saying and how is he saying it? (Note: these questions don't come first! For 99% of the works on display, you don't care about these questions today.)
Before reading the caption, make your own interpretations. Ask what techniques were used to achieve such a powerful effect on you. Take some notes.
You've saved the signage for last. Go ahead, but don't worry about giving the experts' interpretation too much credence. The caption may be interesting, or it may be a worthless detour. It certainly isn't the point of the experience. For some people a museum visit becomes an endless effort to collect stray facts, as if the brain were meant to store them like an encyclopedia. I bet your own notes will end up being the ones you want.
Now you may take a break. Get a coffee. Don't feel like you need to keep going forever. A successful museum visit, in my opinion, includes deliberately limiting your intake. If two or three works grab you in an afternoon, that's plenty. However, while you may disregard (for now) the works which do not grab you, do not use this ignorance-by-choice to pass judgment against them. If a piece is in a museum, you can bet it has true merit, and your inability to know that merit at the moment is surely nothing to brag about.
Read more later. You're still blissfully naive about that piece that grabbed you. It is important, eventually, to know the real answers about subject and meaning. If the real answers differ from your interpretations, you should do the hard work of reconciling the balance sheet. In some cases, after learning more you may end up having a diminished view of that piece which was so powerful when you visited. In most cases, you'll confirm a new passion.
Finally, come back later. All those other pieces which didn't grab you the first time are waiting for their chance to try again!
(Update: By contrast, but making the same essential points, the New York Times says "slow down.")