I remember seeing them in art museums as a kid, but at that time all I really wanted to see in museums was dinosaurs, suits of armor and stuff like that. As I grew older, I had saw the prints and couldn’t figure out what the big deal was. But seeing them is person is spectacular.
As you said, the National Gallery has a wonderful collection, I think the best is probably at the Met in NYC, followed by the Art Institute of Chicago. The collections at the Bellagio and Wynn in Las Vegas are small, but very good.
some JSS showing influences:
http://www.thecityreview.com/f00camp6.jpg
http://www.cassdesign.com/images/panel19.jpg
http://www.tfaoi.com/mn/mic/mic195.jpg
http://imagecache2.allposters.com/images/pic/MCG/pf677_b~Seista-Posters.jpg
http://www.abcgallery.com/S/sargent/sargent6.html