Topeka with a 3-year old means you start at Gage Park, between 6th & 10th Streets on Gage Blvd, right in town. Attractions there include the Topeka Zoo, miniature train rides, a playgound with climbable dinosaurs, lots of woods and picnic areas, and the Blaisdell Pool, which has water play areas - as does the new Shawnee North water park, though I haven't been there. For your dining enjoyment, there's a Chuck E. Cheese on Wanamaker just south of 21st St. You might also check out Discovery Place at the Kansas Museum of History, with kid-friendly exhibits and stuff they can do. There are cheap movies at the theatre just north of the West Ridge Mall on Wanamaker. There are kids' programs throughout the summer at the Topeka Public Library, which is on 10th Street, about a half-mile East of Gage Park, so you could go there before or after the zoo. That's probably too much of an answer, but should be a good start .... I just thought, though, if you are into that sort of thing there are a lot of Vacation Bible Schools in the area every Summer, and that would be a good place for the 3-yr old to meet some playmates in a safe and inexpensive setting.
As for camping, it would kind of depend on where Hubby is staying or working. Lake Shawnee has nice camping spots and is just on the east side of the City. For a little more of a drive, you could go to Perry Lake or Clinton Lake, which both have nice state park and Corps of Engineers campgrounds. For wide-open pull-through type of campgrounds, Capital City RV is on the southwest side, Forbes Landing is by the raceway and airport out south. There is a place off US 24, northeast of town, which used to be a KOA, and is in a quiet, country area with some shade, without being too far out. If it matters how close you are to Hubby's work, let us know. Also, if he's in the area, he could take a quick drive to check out the campgrounds, as there aren't that many of them.