This is one of those things you have to do once in life. The Arch is really fantastic. From a distance, you'll think it's too narrow for people to go up to the top--you'll still wonder about that even when you're right up close. But you can ride up, in claustrophobic elevators; though not a long ride, you should allow 45 minutes because you'll have to wait in line before getting on. The best way to visit is by buying a ticket for a specific departure time and then showing up for that time.
Once up top, you'll get great views of the city, the midwestern plains beyond, and the river. Once, when I was young, I went up with my school class and about 10 to 15 of us ran from side to side in unision until the whole thing started to sway and we got in trouble. A windy day will have the same effect, which can be a little unnerving. It is, however, completely safe up top, even if doesn't always feel like it.
Below the Arch is the very good Museum of Westward Expansion that highlights the Lewis and Clark expedition. Here you'll learn about life on the frontier, Native Americans, buffalo, etc, for free, which is a good way to pass the time while waiting to go up. For a bit of money, you can watch movies about the Lewis & Clark expedition and the construction of the Arch itself ($6 for one/$10 for both). They're good enough, but not necessarily worth making a point to see.
There's also an IMAX theatre in the building as well.
First and last Arch rides are at 8:30am & 9:20pm during the summer, 9:30am & 5:20pm the rest of the year. Cost is $7.