I would back up Janek's Statement about the excellent public transport, and that kids under 6 are entirely free.
The transport company is called BVG, so you can look up transport connections at bvg.de (in English too) a excellent resource. For your own ticket, it's worthwhile getting a flat-rate ticket for your entire stay (for 3/4/5 days) - there are Kiosks at both airports so you can buy them at when you arrive. If you arrive at Schönefeld DO get the ABC zone ticket, not just AB.
Hmm, what else? Oh yes, a law has been passed against privately renting out entire apartments (so an airBnB flat is illegal, but a room in a flat is ok). There are still lots of flats on offer, but do you want to take the risk?
MachMit Museum - Great idea.
Once of our favourite things with our kids at that age was going to small kids' theatre performances, often on weekends at 4 pm all over the city, but most of them will be in German. But if you find one in English, that is the most 'authentic' family activity in these parts.
Efficient transport - compared with the UK, definitely. I once apologized to visiting friends because their underground train arrived two minutes late, they thought it was hilarious how German I'd become!