Visit to Germany in June

Skip the Black Forest from Stuttgart. The Black Forest has name-recognition, but it's just one of about 50 different low mountain ranges spread across southern and central Germany. The only impressive part is the far south, which is a few hours from Stuttgart. Not worth the effort. If you want to see a rural, forested mountain range, the Schwäbischen Alb arise just to the the south and east of Stuttgart. You will find plenty of castles here, including the impressive Burg Hohenzollern, near Hechingen.

If people in your family have an interest in science, I'd highly recommend the science museum (Deutches Museum) in Munich. It's outstanding and in a class of its own. Google it to see if it is something your family would enjoy. I would agree with Tom' assessment of the over hype of the Black Forest as a long day trip, especially with kids.

I presume you are flying into Berlin. Are you flying out of Stuttgart? If so, I would cut a day from Stuttgart and add it to Munich to make up for the day in transit from Berlin to Munich. If departing from Frankfurt or another city, I would spend the final two or three days there. Nothing against Stuttgart, but Bavaria and the Rhein-Mosel area have much more to offer a first-time visitor.

As a second to Roy's comment, note that the Cologne Cathedral is roughly tied with the Reichstag in Berlin as the most visited monument in Germany. Cologne has a lot to offer, and regionally as well. (I should admit that I'm not that interested in fine automobiles.) The Black Forest is indeed pretty, but driving (once you get there) is time consuming on curved and hilly roads. You also need time to get out and hike and visit - otherwise it's just a green forest.

I agree about the Deutsche Museum and Dachau as important and interesting sites. I found Neuschwanstein to be a waste of time. Nymphenburg Palace and the downtown Royal Apartments in Munich are must see's, however. Stuttgart is about 140 miles west of Munich, and the Black Forest is west of there.
Spending the time 2 hrs. south of Munich in the Austrian Alps would be much more pleasurable and it is an incredibly beautiful area.

I have to disagree strongly with Tim about spending time in Cologne versus Stuttgart. I'm not heavily biased towards Stuttgart as a travel destination itself, but it's a far lovelier and more enjoyable city than Cologne, which aside from the fischmarkt area was maybe the most charmless city I've been to in Germany. All my friends I've spoken to felt the same way about it. I'd recommend controversial Stuttgart or Frankfurt well over Cologne, ESPECIALLY for kids. There's a lot more family friendly stuff to do in Stuttgart than Cologne. That said, 3 days in Stuttgart is a little much. Here's what I'd do: Leaving the royal castles (do that your last day, driving) head to Lindau to overnight. It's on the Bodensee (Lake Constance) and there's a lot for young people in the area, like water parks. It's lovely there, and you'll be passing great Alpine scenery from Fuessen to Lindau. From there, head around the lake and up towards the Swabisch Alb, either to Hohenzollern castle, or Tubingen (or both). Tubingen is a beautifully preserved old Swabian town. You can do a combo of scenic back road driving or stick to the A81 if you want to get somewhere fast. Stuttgart is only 30 min from Tubingen. Then I'd spend two nights in Stuttgart. Definitely do the Mercedes museum - best car museum in Germany - and Porsche only if you're really into Porsches. Depending on your son's ages I'd consider the Wilhelma Botanical Garden and Zoo - way cooler than it sounds, it was created by a King of Wurttemberg and has very unique and beautiful architecture and some great animal exhibits, especially for primates. The newly revamped State museum of Wurttemberg in the Old Castle is also super cool with lots of Roman and Celtic artifacts.