Purchasing train tickets

Getting a seat is not a problem. It's how much you want to pay for it. You can always buy tickets immediately before the departure of the train you want, but you'll pay full price. The reason to order in advanced is that it often gives you a significant discount, although this can lock you into taking a speciftic train. Explore the Deutsche Bahn website for more information.

Adding to the excellent information Tom gave you:

Here's Rick's tutorial on using the Bahn website: http://www.ricksteves.com/plan/tips/db_tips.htm

Note that the Bahn website will only have prices for trains that start or end in Germany. And yes, if you can commit to non-exchangeable and non-refundable tickets (or tickets that have severe restrictions on these actions), you can save a lot. When my mother and I went from Berlin to Dresden, a full priced ticket was €38 per person, a discount ticket for one was €19, and a discount ticket for the two of us traveling together was €29. So, we saved €94 on that round-trip!

To look at ticket prices for trains starting in Austria, use http://www.oebb.at/en/index.jsp. For the Czech Republic, use http://www.cd.cz/en/default.htm. A quick way to find out if there are discounts for advance booking: look at trains for tomorrow and then for you travel dates. When looking, be careful of the times; most websites use the current time in Europe as their default, which is often much later in the day than you want to travel.

Only express trains in Germany (ICE/IC/EC) have reserved seats. Seating in regional trains (IRE/RE/RB) is first come, first served. Whether you are using advance purchase discounted tickets, paying full fare, or using a rail pass, you can reserve seats on express trains, up to 2 trains in one direction for 4,50€/seat, 9,00€ for a family of up to five.

If you are going by regional trains, you might as well buy your ticket at the station just before travel time. As long as you are traveling after 9 AM workdays (anytime weekends), you can probably save by buying regional passes like the Bayern-Ticket in Bavaria. As long as there is at least standing room, they will let you on board.

If you are going longer distances and want to use at least one express train on the ticket, you can get advance purchase "Sparpreis" ticket starting at 29€/person. These tickets must be purchase at least 3 days in advance (up to 92 days) and are tiered at each price level. When tickets at that tier sell out, the price goes up.They are purchasable online from the Bahn before you leave and printable at home. These ticket are train-specific and non-refundable as of the travel day.

You can also do the math to decide whether you want a railpass for one or two of those countries, or for four (Select pass or European East). This may not add up to a savings, but it depends how far you're traveling, especially within Germany, which has the longest distances and ticket prices can add up.