What is the cheapest way to get to Gottingen from Berlin ? I checked the Bahn Website and found some routes from Tegel airport. Are there any other cheaper routes ?
I also found something called "Savings fare". What is it ?
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Train is the fastest and bus is the cheapest means of transportation for that route.
For a departure in seven days I found the following:
For a departure today, there are still bus tickets available at 18 EUR. The train fare is 80 EUR in that case. If you are fine with travelling on local and regional trains, the fare drops to 44 EUR ("Quer-durchs-Land-Ticket"). But the travel time goes up to 5h00 - 5h30.
If your final destination within Berlin is Tegel Airport, you have to add 2.70 EUR to the bus ticket and 1.60 EUR to the train ticket, for local transport.
The 29 EUR train ticket is a "saving fare" (Sparpreis). These tickets are only available in limited number. The ticket has to be booked at least one day before the departure.
Saving fares have some particular cancellation restrictions, just as long-distance bus tickets have.
Have a look at Mitfahrgelegenheit. Its search will show you "Sparpreis" fares on the Bahn, buses, and people offering rides in private cars in exchange for gas money.
You might also want to look at l'tur. The "Bahnreisen" section has cheap connections for one country every month. If that happens to be Switzerland, you can get a direct ICE to Basel that stops at Göttingen.
The deal with Sparpreis tickets is that you have to go on the named train and can only partially refund them before the day of travel. But they can be quite a bit cheaper.
I also checked in on the Bahn.de site and found that it is a 2 hour plus journey, even by fast train. That means that there is likely no local buses that can make your journey fast and cheap, there are long distance buses and you will find more lower down this answer.
The standard cost of the tickets is €80. The savings fare means that you book your ticket for one particular train, you have to travel on that train or have to buy a new ticket for an other train. But the tickets are often much cheaper, for coming Monday you can get them for less than €60 and when booking further ahead you can get them as cheaper, I have seen them as low as €29. According to one of the comments they are refundable but expect to pay a high fee for that. Do enter your age into the planner if asked, as for some travels (surely international ones) it can make a difference.
If you have to start from Tegel Airport you have to travel for a bit by bus, making the journey a bit more involved than when you start from a railway station. But that site will give you all details for the travel needed. When there is no reason for a start from the airport, I would start from the main railway station, the Berlin Hauptbahnhof.
A quick google search showed up several bus search sites, I used this bus search site and if found regular buses from Berlin to Göttingen. In the results it also shows train journeys, so check out that you do look at the bus journeys. They seem to start from €18. I have no experience with long distance buses in Germany but I expect an early booking low price and a higher price when you leave buying tickets to the day you travel.