Is SSL broken? If so, what can I do about it? But recently, with the scandals about the (in)filtrations, I read that the NSA (and of course others, but I'm not particularly worried about NSA) is able to break SSL and track your data.

It is true? What can I do about that, given I do not know much about cryptography?
I don't think NSA can break the underlying encryption primitives. What they may do is record the whole SSL traffic and decrypt the whole traffic using server private key. 
So you can use introduce perfect forward security as the blog article “SSL/TLS &amp; Perfect Forward Secrecy” suggests. 
In addition man in the middle can be performed with fake certificates. This is relatively easy and not only NSA, but also a hacker can do this. So if you consider other parties except NSA you should look in to this.