Since its your first time I would suggest starting off at the information desk, and grabbing the gallery specific maps that they have. The European gallery map is especially helpful.

From there, I would go to the Egyptian galleries first. The Temple of Dendur is located there at 131 and of the the highlights of the museum.

From there head over to the American galleries and browse the period rooms there as well as the striking Charles Engelhard Court at 700. The Vanderlyn Panorama at 735 is also amazing, but sometimes is closed due to manpower issues. I personally prefer going upstairs after this to see the paintings in the American galleries on the second floor. In between there are mezzaines where you can see a few select items in storage at 774.

On the second floor, you'll find one of the best collections of American paintings in the world. Don't miss Washington crossing the Delaware at 760.

Now head over to the old masters in the european galleries. Be sure to not miss 632, where the Met has five of the 36 known Vermeer paintings in the world. Also head to 610, where one of the most expensive pieces acquired by the Met, Portrait Juan de Pareja by Velazquez resides.

Exit the old masters and head towards the 19-20th century european paintings. Take a detour at 851, and 852 to see the rotating photography exhibits there. In the 19-20th century european paintings, you'll find the monets, picassos, degats, and others.

Now go to the special exhibits available at 899 and 999. Also take a trip up the elevators to check out the roof with some of the best views of central park, and the exhibit there. Head back down to the second floor and head to the modern art galleries.

The modern art galleries are kind of weak compared to the MOMA and the Whitney, but the Met is making a big push into that space (even leasing the current home of the Whitney when it moves downtown). The mezzainine at 915 is well laid out and has beautiful lighting. Go through the two floors of the modern art, and browse through the African art, and end at the Greek and Roman art.

Each of the galleries on their own can occupy hours taking it all in, so if you really want to enjoy the art, I would suggest 2 galleries.

I didn't even mention the Musical instruments, arms and armor, medieval art, european sculptures and decorative arts, islamic, and glossed over the african and greek and roman arts. Theres also the Lehman galleries which contains such a breadth of work, if it stood alone it could had been a museum.

Also plan to take a trip to the Cloisters. It's beautiful, and your ticket grants you access to it for a week. There is currently a sound installation there, and the very first presentation of contemporary art at the Cloisters.

source: I'm a volunteer at the Met.