I am planning to travel from Cologne (Köln) to Prague. Some one suggested me to search for some cheap flight. But, I want to avoid this. What are the other best possibilities to travel from Köln to Prague (best means cheap and time saving)?
If it has to be simultaneously cheap and time-saving, it will be hard to beat the plane.
You may consider a trip by bus. It takes roughly 10 hours and is probably cheaper than the plane if you plan on a short notice. See e.g. here: https://www.busradar.com/
10 hours is also the duration of a day-time train, via Berlin. There are combined train/bus routes, via Nürnberg or Mannheim, sold by the Deutsche Bahn. Travel time is a bit less than 8 hours via Nürnberg and approximately 10 hours via Mannheim. If you book in advance and/or on less busy dates, these options can be quite cost effective.
There is also a direct night train from Cologne to Prague. It is really time effective. You travel while you sleep. And if you book sufficiently in advance it will be pretty cheap (also considering the fact that the train ticket includes the accommodation).
The fastest option is to take an ICE train to Nuremberg and travel onward by bus from there. Total travel time is around 7.5 hours. The ICE trains as well as most buses between Nuremberg and Prague are sold through Bahn.de, if you book well in advance you can get tickets for around €40, buying on departure at the station can cost upwards of €150. There is also a slower direct overnight train with similar pricing.
There several bus companies that offer direct buses from Cologne to Prague for around €30 to €60 euros regardless of time of booking. At least four different bus companies travel overnight and one by day. Travel time is 10-12 hours. Busbud lists departures and prices (full disclosure: I work on this project).
If you don't mind the organizational work, carpooling is an option, BlaBlaCar and Carpooling.co.uk are the main sites.
You have got some good answers already but if you want to explore what's possible with the train, one nifty tool is the DB “Sparpreis finder”. It's basically another interface to their booking engine but instead of specifying a time and seeing a handful of detailed connections ordered chronologically, you can specify a date and see all connections ordered by fare. You can also navigate easily from one day to the next. It's particularly useful if you are somewhat flexible regarding the date and want to find the cheapest discount fares on your route.
NB: My link is in German, I don't know if it's available in English. Most options should be self-explaining, the only somewhat counterintuitive thing is that to search trains to Prague, you first need to click “Start oder Ziel außerhalb Deutschlands” (point of departure or arrival outside Germany”), then leave “Ziel außerhalb Deutschlands” (destination outside Germany”) selected and choose “Tschechische Republik (CZE)”. When you come back to the main screen, the destination field will have been replaced with a list of Czech train stations.