Housed in a neoclassical building overlooking Trafalgar Square, the National Gallery boasts greatest collection of European paintings in the English-speaking world and has always seemed to me the very paragon of what an art museum should be. It's grand, centrally located, lavishly decorated with marble and dark wood, and despite modern concessions to accessibility, remains redolent with the passage of time in a manner that has greatly enhanced my experience on every visit there. After climbing the stairs from the sterile modern lobby and opening the heavy doors to the grandly decorated Central Hall, I've always felt cultured just by association.
The collection spans the years 1250-1900 (later works are in the Tate Modern across the river in Southwark), and is unusual for a National Gallery in that it primarily contains foreign works. This is due to the creation of the Tate Britain (in Milbank) for the specific purpose of displaying the bulk of the British works in the collection; but unless you're particularly interested in the work of J.M.W. Turner (whose works comprise a wing of the Tate), you're better off here as the average quality of British works on display is much higher, with John Constable, Thomas Gainsborough, Sir Joshua Reynolds, and Turner all represented by much of their best work.
The Gallery's Continental collections, much (but not all) of which are arranged by country as well as era, are the source of its reputation and breathtakingly comprehensive. From anonymous medieval Italian devotional paintings to Vincent van Gogh's "Sunflowers", they're an art-history book brought to life. Indeed, handily for the majority of the visitors who prefer to hit the highlights, the complimentary Gallery map features pictures of the most famous paintings matched to the rooms in which they're displayed. Although this approach is better than nothing, it prevents you from appreciating the depth of the Gallery's collection (all of which is permanently on display) particularly with respect to individual artists and time-periods, as well as its many diamonds in the rough.
You can sample the collection in a long morning or afternoon, or spend a day combining it with the next-door (and frankly disappointing) National Portrait Gallery but really the more time you spend here, the greater the reward. It was only on my last visit that I came to appreciate the extent and variety of its collection of Raphaels, which are housed along with the other pre-1500 works in the Sainsbury Wing. While there, you shouldn't miss Leonardo da Vinci's "Virgin of the Rocks",Jan van Eyck's "Arnolfini Portrait," or Michelangelo's wonderful unfinished "Entombment." The options in the North Wing are even better; you could happily spend an hour basking in a room or two alone, particularly the extensive collections of Rembrandt and Constable. Personally, my favorite painting here is Caravaggio's early "Supper at Emmaus."
Don't just follow my recommendations (or any others); the National Gallery is best explored on your own! The temporary exhibitions tend to be excellent – and expensive.
http://www.nationalgallery.org.uk