How to find out which class I'm currently in in Pycharm?

Try ALT+Q, or Mac: Ctrl + Shift + Q for Mac as described in the PyCharm docs

Navigate | Select In... | Project View or File Structure, or you can keep Project or Structure view visible with the Autoscroll from Source option enabled.

It's also possible to use Navigate | File Structure for a pop-up showing your current position.

Breadcrumb bar just under editor

As of Pycharm 2017.02, there is a breadcrumb navigation under every editor window that shows information like:

ClassName > FunctionName > for i in range(): > if True:

You can also click to go to any of those points in the code.