Barb: I've bought advance-purchase savings-fare tickets at the DB website several times. You can buy tickets for within Germany or between a German city and a foreign city. They are quite good as long as...
1.) ...you can commit to keeping a strict travel schedule (tickets are train-specific and not refundable/reusable at the last minute.)
2.) ...you are traveling fairly long distances using high-speed trains (IC, ICE, EC) for at least part of the journey (local/regional journeys that use only local/regional trains are not part of the savings fare program.)
3.) ...you are able to buy tickets 3 months in advance (if you postpone purchase, prices rise as the tickets are sold to others.)
If your trip is typical, you will also have some local/regional-train-only travel. These trains have a variety of designations including S, RB, RE, IRE, MRB, and many others. Tickets for these trains are sometimes sold by DB, and sometimes not (NOT if your journey is within a local travel zone, where the local transit authority sells them instead.) But there is no reason to purchase any of these tickets in advance. There's no discount for advance purchase, and they are sold at all stations via ticket machines. No seat reservations are possible, and tickets never sell out.
Whatever discounts might be available for these local/regional trains are available at the ticket machines on the day of travel. And the discounts tend to be substantial. Daypasses from DB and from local transit authorities offer huge discounts. Here is a sampling:
Bayern ticket (one of the Länder Tickets)
VRM tickets (local transit authority in the Rhine/Mosel region)
You can save a lot by purchasing carefully, but how to get the right deal for your trip isn't always transparent. Best thing to do might be to post your itinerary for ticketing suggestions.