This is one world-class museum nobody can miss! And it's the best price in the world, FREE! Even the kids will like it! If you are without children on the visit, plan to spend at least half a day there, as it is so huge, it just takes that long to see the important stuff. I recommend taking an afternoon to see it. Tours and school groups tend to come in the mornings and that makes the museum crowded. Crowds make me feel overwhelmed and often the more important pieces are hard to see.
If you chose to go, you must see the Egyptian collection, the Greek and Roman collection (which includes the Elgin Marbles -- the Parthenon Friezes), and the Near Eastern sections. The bulk of these are on the first floor of the museum and are often jumbled up together. The Near Eastern collection includes a great deal of artwork from ancient Iraq and the section is almost always empty. The museum can be fickle about letting people in that section, especially if its crowded (the space is rather narrow), but be persistent. Slightly out of the way on the first floor is a collection from the Mausoleum of Hallicarnasus, an important Greek monument from Turkey. Don't miss it – it’s cool and you won't find as many tourists there as in the neighboring Elgin marbles. Also don't miss the Egyptian mummies on the second floor! The mummies are almost always crowded, but well worth the agony! If you have time, see the Roman Britain section on the second floor.
The new courtyard in the center of the museum is a great place to relax and have a cup of coffee and tea, too.