I would find out in advance if there is a really major work or some really major works that I want to see - ie. by one of my favourite artists who I've already seen a lot by, and I would seek those out and prioritize them. I would linger around those works, and probably visit other areas of the museum a bit more rapidly. If there is nothing specific that I am really dying to see, I would concentrate more on a general area I'm interested in (ie. Dutch Golden Age painting, or whatever).
In terms of my response: I was fortunate to have parents who really appreciate art and know quite a lot about it. So I've had a chance to build up some knowledge (though I have no visual artistic skills myself.) My response is a definite intellectual/emotional combination. Part of me may be comparing the work to others I've seen by the artist, deciding if it's typical or untypical, how much I like it in comparison to the others, etc. Part of me is just having a sort of wondering emotional reaction. It's hard to put into words sometimes what kind of emotional reaction I'm having, but one way I can tell that it was strong is if I look at the work for a while, go away and look at other areas of the gallery, and then feel drawn to come back and linger by that work again - even feeling reluctant to leave.
I've occasionally been motivated to go away and write a poem or something, but my poems on works of art haven't been all that successful so far, I think...
I love art galleries, particularly if they swing toward the type of art I love. I'm not a great appreciator of modern art, but it helps an awful lot if I go with someone who can tell me a bit more about the meaning and the technique behind the works. I've had that experience with friends, especially artist friends, who are much more knowledgeable in the modern art area. It really opens things up. It also confirms my belief that you can love and appreciate art without knowing a lot about it and certainly without being an artist yourself, but I do believe that if you have some knowledge of the artists, their milieu, their intentions, even the technique (something I have an extremely hard time grasping - in terms of materials and so forth, I have no grasp of that), you will find more to appreciate. I believe this to be true of all forms of art, not just the visual arts - ie. also music, literature, etc. I guess that's why my reaction is always partly emotional and partly intellectual.
I tend to feel peaceful and happy in art galleries. I will have an especially happy time if it's a gallery I've never visited before, but I know there are works in it I'm eager to see, by my favourite artists. Or if it's a gallery I've visited many times but I'm going to spend a few minutes with one of my fav paintings of all time. It's a great feeling. I have a similar feeling when I go to classical music concerts. It's like a hole fills up inside, for a while.