Yes you can, but you might not get the best deals.
For buying tickets from Germany this is the "official" website:
http://bahn.de
If you use that site above (there is an English version), when you see things such as "Berlin Hbf", it is short for "Hauptbahnhof" which means "Main train station".
There are several train stations from every city. You can use google maps to see if their location is more convenient.
There is a cheaper alternative to trains: car pooling : mitfahrgelegenheit.de
There is a Museum of technology in Berlin. It is not free.
http://sdtb.de/English.122.0.html
There is the museum of Military aviation a little outside berlin - it is free. Not to miss if you're interested in this stuff:
http://en.wikipedia.org/wiki/Luftwaffenmuseum_der_Bundeswehr
It is on the outskirt of berlin. You will need a zone C ticket. Let me know if you want to go and you need help finding it, because their web presence is utter crap.
Otherwise check out the Tempelhof park which is a former airport which was converted in a huge park. Converted is not the right word... it was basically opened to the public and renamed "Tempelhof Freiheit Park".
I also recommend Burgermeister (but you have a 3/4 hour wait if you go on weekends), and check out the Kottbuser Tör area and it's pubs and bars.
And of course there's the Museum-Insel (Museum Island) which is not free but worth the investment if you're into museums.
I guess your missus and you will love an excursion to Potsdam (zone C, again) , to look at the former imperial palaces there.
That's not "geeky" stuff, but there are a lot of things to see in Berlin, and 3 days is short. I tried to recommend the best stuff.
A zone C ticket is a transport ticket (metro, bus, etc.) that allows you to go far outside Berlin. It is not expensive, but the trip can take 1 hour, or even more, depending on where you start.
Also see if there are any concerts or events you're interested in during that period in Berlin. There are a lot of these happening there.
And oh, of course, there's Checkpoint Charlie and the wall (or Mauerpark), quite touristy stuff but again worth seeing.
There's also Grünewald (a small forest) and the abandoned former american radar station (now covered in graffitis) in the middle of it. Last time I went to that area there were a bunch of Israeli "guides" demanding some payment to visit the abandoned station.
PS: There's a "welcome berlin" card for like 3 or 4 days, it gives you unlimited public transport and substantial reductions on museums and many attractions (with a little book listing them all) - but it's only worth it if you're planning to spend some money on commercial attractions and museums, otherwise 3 or 4 daily tickets will cost less.