If you would just like to stand on the equator, your best bet would be to visit Kenya or Uganda. And while you're there, be sure to check the flushing toilet theory, or to be more scientific, the Coriolis effect.
Toilets flushed in the northern hemisphere apparently spin to the right, in the southern hemisphere the water spins left -- this is supposedly caused by the Coriolis effect. And on the equator? It's a straight shot down. You can test this theory if you stay at the Mount Kenya Safari Club where you cross the equator just by going from your room to the dining hall.
One of the nerdy joys of standing on the actual equator is not looking silly while conducting your own scientific experiment. Fellow traveler Dan Greenspan conducted his experiment using a simple bowl with small hole in the bottom, some water and grass. His water spun to the right when he stepped into the northern hemisphere, to the left in the southern hemisphere, and headed straight down when he was directly on the equator. Thus refuting a BBC report that the Coriolis effect is impossible to measure on such small scale.