Touring Character Shows
For other Disney shows like Disney Jr. Live and Phineas and Ferb Live, etc, go to Disney Live.
This was a lot of fun and very educational. Anyone who is interested in dinosaurs would love this event.
Nickelodeon Touring Shows
The Fresh Beat Band, pictured below, was Nickelodeon's kid show about a group of singing kids! Their tour, complete with singing and dancing was great fun to watch.
While The Fresh Beat Band was a lot of fun, their day has passed. Kids shows only have a typical run of 2 to 4 years. However, Nickelodeon generally keeps some sort of yearly tour going with its shows.
This is the first year for this, but with how amazing and fun it is, I can't imagine it wouldn't continue, especially since the world (and my boys) have gone superhero crazy in the last few years!
Musicals
There are a number of musicals that tour the country, many of which are family-friendly. From School of Rock and Spongebob to countless Disney adapted shows like Lion King and Aladdin, there are plenty from which to choose.
The content of the show changes yearly with all your favorite characters joining into each new plot.
Thomas tours the country with the event called "A Day Out with Thomas" which includes a ride on Thomas, some of the characters from the show walking around and multiple other activities.
Yo Gabba Gabba, Wild Kratts, and whatever the kid show dujour is usually have a show touring the country, coinciding with their popularity. These shows are your chance to see the lovable characters up close and in person.