We are thinking of taking advantage of that cheap fare to Cologne in April ($266 from Newark roundtrip). Is there enough to keep us entertained in Cologne for 6 days? Is it easy to take day trips to any other cities? Any suggestions?

I had a very full day in Cologne and loved it. If you go at a more normal pace, you could stretch it to 2-3 days. There are nearby towns like Mainz and Koblenz that are meant to be nice, though I've never been. You could also visit Bonn. I think you will find plenty there and around to keep you busy. It is also a good transportation hub, so you could train to other parts of the country as well.

Thanks JoeTro!

Cologne has excellent museums, a lot of other interesting churches in addition to the cathedral, great nightlife and shopping. Besides, Cologne is a good base for getting around by train. No need to bother with a car.
There is more than enough to keep you busy. (Google for the places mentioned):

- The Middle Rhine Valley: train to Bingen, then back by boat to Koblenz, return to Cologne by train.
- Aachen
- Düsseldorf
- Brühl with the lovely palace Augustusburg
- Bonn, as mentioned. If you like hiking, cross the Rhein there and walk up the Drachenfels, a castle ruin on top of a steep hill with great view

What keeps you entertained?
I'm not sure the Koln itself would be enough for a week but the surrounding areas certainly would.

Berlin is les than 5 hours by rail; Mainz and the Rhine valley less than 2 hours; there are high-speed connections to Brussels; possibilities are endless IMO.