The shrine is at the center of this large park. There's a small coffee shop at the entrance, a gift shop nearby, vending machines over the far side by the treasure museum, and a pay-per-view inner garden which is pretty enough at the right time of year (around June). There are also really loud loudspeaker announcements several times per hour in English and Japanese pointing out that it's imperial property, so don't attempt to drink in the wrong place, smoke at all or throw balls on the grass. What would Japan be without loudspeakers?
There are two areas to the shrine: the inner precinct, or Naien, and the outer precinct, or Gaien. The outer precinct is in a completely different place and is actually best reached via Gaien-mae Station or Aoyama 1-chome Station on the Ginza Line. It consists of the Meiji Memorial Museum and several large sporting facilities. Whilst the inner garden is worth a visit, the outer garden isn't. Don't get confused: The outer precinct, or Gaien, is completely separate from the Naien where the shrine is. The pay-per-view garden inside the Naien is called the Gyoen.
The original Meiji Jingu burnt down during WW2, and most of the current buildings date from 1958. The wood is mostly Japanese cypress and the roofs are copper. On entering the shrine itself, the hall to the right is the Kaguraden, or Hall of Shinto Music. It was completed in October 1993. It's much larger than it looks: it has only one floor above ground, so as not to dominate the skyline, but two large floors underground. The main hall here can seat 800.
If you collect temple calligraphy, this is where you must go. There is a small reception desk on the right, where you can hand over your book. The blindingly obvious place in the temple itself, selling good luck charms and temple stamps, is the wrong place... (Early 2004 update: a new building has been built to the right of the shrine to sell charms, etc.)
Meiji Jingu is a popular place for weddings. There is a continuous stream of wedding processions each Saturday and Sunday. There are also many festivals. One of the nicest is held in mid-October, and is called Ningyo Kanshasai (Festival to Thank Dolls). People all over Japan send dolls here when the dolls have outlived their usefulness as children's toys. The dolls are displayed around the courtyard, and a ceremony is held to bless them for their work. The dolls are then disposed of. This is a good festival for the casual visitor: unique, yet not too many people.
Japanese festivals tend to be held by fixed date, rather than the nearest Saturday/Sunday to a certain date, as in many Western countries. So, April 29-May 3 is the Spring Grand Festival, with displays of traditional dance (Bugaku), theatre (Noh), music (Sankyoku and Hogaku) and archery (Kyudo). This is repeated during the Autumn Grand Festival with the addition of horseback archery (Yabusame), martial arts and Aikido. Of course, the area gets stuffed with people, so you don't actually see anything.
The Gyoen is best reached via a small entrance on the left of the main path leading to the shrine itself. It was created by combining the lands of Daimyos Kato and Ii. The main feature is the iris garden which was designed by Emperor Meiji.
After looking around the shrine, if you have time it's worth heading out through the left exit (as facing the shrine itself), away from Harajuku. The follow the paths to the right (either will do) towards the Treasure Muesum. You'll come across a relatively empty grassy area. This and the grass in Shinjuku Park are some of the few areas in Tokyo where you can actually find a large patch of decent grass and sit on it. The Treasure Museum doesn't contain much treasure as such; it houses mostly items used by Emperor Meiji and Empress Shoken. It dates from 1921 and is made of ferro-concrete. It's open only at weekends and on public holidays. Worth a visit only if you have nothing else to do. (2004 update: it now appears to be open every day except Monday, unless Monday is a public holiday in which it's closed on Tuesday)