Under the assumption that you will use some kind of design resembling the picture you posted, especially the diagonal braces on the ends, I would use two 2x6 for each side rail.
Capable of supporting a 50-75 lb child swinging in the middle (6 feet from either end). And possibly multiple kids if the neighbors invade the backyard.
I think the lumber will lose most of it's stability with the holes that are drilled for the steps/rungs. To compensate for that I would drill all the way through one 2x6 and glue a second one to it. This has the added advantage that you could clamp the two rails with the holes together while drilling, ensuring the holes to be nicely aligned.
One other crucial part regarding the stability is the selection of your lumber. Be sure to avoid posts with as little knots (if any) as possible, they are predetermined breaking points.
Douglas Fir since this will be outside in the midwest
We had a bench built with untreated Douglas Fir and after 4 years it splintered all over the place. I heard them say this was because it was "untreated". Something you might want to consider when building something for children.
Not excessively over engineered so I can still move it around if necessary. For example, if I don't need to use 2" x 12" boards, then I won't.
I don't see a way to avoid the long boards, but all in all I think two moderately strong people should be able to at least carry it around the backyard a bit. The ladder in your picture looks as it could be disassembled fairly easy, which probably would add to the task of engineering it though :)
Another thing to consider is, if it's easy to move, it's also lightweight and prune to tip/collapse. A couple children playing with it might tip it over and hurt themselves if the thing is too flimsy.
My grandfather built something alike for us and we kept messing around on/under/with it way longer than he anticipated. It basically had to hold the equivalent of 3 fully grown men in the end :). So you might want to consider dimensioning the thing to hold 200 lb (~90 kg) in the middle.
Out of curiosity, what are you planning on using for the steps/rungs? I'd go with quarter or half inch pipes I think.