We purchased our tickets online and way in advance. This was mostly because everyone and their brother wanted to get in during the first few days of it opening, which I completely understand. You may a very small fee, like two dollars a ticket, when you would usually visit completely free. Visiting the monument during the summer without purchasing before hand could go either way. If you're looking to get in for free I would show up early to the gift shop because they are first come, first serve. I personally like the convenience of purchasing online for the exact date and time I want. The visit is not a long one, but you have to show up early to get in line because they go by groups and load you into the monuments "waiting room" of sorts, near the elevators inside. The kids loved it, even Phillip. You look out the windows, there are old pieces of the monument to touch and various other historical things to read and check out. On the way back down you get to check out all the donated placemarkers that line the inside of the monument. There's also a "Stair Way Tour" you can take, which is on Mr. Air Force's and my bucket list.
If you are going with kids I definitely recommend picking up the Junior Ranger booklet. We try to get these wherever we go. It provides an education experience and the kids usually get a patch or pin marking the occasion.
For more information on the Washington Monument visit the National Park Service's website on it or leave questions here for me to answer.
If you are looking for tips for being a tourist with kids, check out my Five Tips post, here.