Hi everybody!
I will try to answer your questions concerning Köln (Cologne) and Frankfurt.
The christmas markets in the cities are really worth a visit - and they will keep you busy Amer:) I guess if you will mainly stay in the Frankfurt area and don't buy half the christmas market, then 300 euro should be fine - the public transport isn't too expensive and there are cheaper 3 or 5 day tickets. I guess in Frankfurt it can be a bit hard to find cheap accommodation, but there are some good hostels...I will look for a link in the next days, but there should be something online under "Jugendherberge Frankfurt"...
It's good to hear that you are interested in seeing more of Germany, Karthik. In Cologne, you can see the most important sights in about 2-3 days, if you're not taking long breaks;) There is a famous cathedral next to the centre, so you can easily connect these sights. Then there are many museums in Cologne (Roman times, art, chocolate,...). This link can give you more information in English:
http://www.koeln.de/en/index.htmlFestivals mainly take place in summer, but there are concerts and fairs in winter. The carnival takes place in february or march here and then the people go crazy and the city is very crowded - as is the city centre each saturday;) But usually Michael is right, the city isn't to crowded on a normal day. If the weather isn't too bad you can also do a river tour on the Rhine and look at some important buildings from the boat. There are also some interesteing cities and landscapes near Cologne, but that would mean a longer trip.
I hope this reply isn't to long and confusing - I tried to answer everybody. If I missed a point let me know and I'll try my best!