Long-term Berliner here. Many of the tips above are just fab, definitely to be followed. I would just add:
Circus is an upmarket hostel rather than a hotel, but great location. It has a sister hostel called Heart of Gold near Oranienburger Straße. The hotel I've always wanted to see is the one that has individually designed rooms, but can't remember the name. It has some really wacky designs - one room is a caravan.
Book your Reichstag visit as soon as poss - the slots go quickly. And remember to take your passports with you cos your ID will be checked.
My experience of Amrit has not been great, but Indian food generally in Germany is dire. Why do it when Indian food in the Uk is much better. The iconic Berlin food is the Currywurst, and the traditional best outdoor place to eat it is Konnopkes Imbiss at schönhauser Allee in Prenzlauer Berg. A Flammkuchen might be nice, but it's traditional for Alsace, not Berlin.
Transport: it's well worth getting a 4-day travel ticket each, because it also gives you reductions on museums, restaurants etc. if you're flying into Schönefeld there's a local transport desk there with people who can advise you, give you a map and sell you the ticket. The local transport network is the BVG and its website (in English too) can work out how to get you from A to B, including walking time.
Re. Buggy-friendly, yes, pretty good on the whole. You can always get on buses with pushchairs (unless there are already two wheelchairs, of course, which would take priority). Most stations have lifts, and they mostly even work.
New buildings by law have to have a lift (disability rights) but older apartment blocks won't have one, and getting to the top floor with a toddler in tow is miserable (bitter experience), so do check your hotel has a lift. Of course, AirBandB is now illegal in Berlin, but I'm sure flats are still being let.
I'm terrible at remembering names of nice cafes etc, but off the top of my head:
Tdjakische Teestube - Great experience of Russian tea house, sitting on carpets etc.
Nola's am Weinbergsweg - Swiss Style but in cool location with nice view.
Anita Wronski - in Prenzlauer Berg, nice pub
Gugelhupf - the place where Bill Clinton ate, but good food as well.
Café Cinema - one of my favourite smoky dark cafes in Mitte - I first went there in 1991 and it hasn't changed since, right on Hackescher Markt. Except it's not smoky now because of the smoking ban, but it feels smoky still.
And finally, can I also recommend the Humboldt Box, with the new Schloss being built behind it. The exhibitions there are being coordinated by Neil MacGregor and are rather good (must confess a professional interest here). It's all free and just opposite the Cathedral /Museum Island so not out of the way at all. Fab view from the top floor cafe of Unter den Linden from the front and the palace construction at the back.
Do have fun!