I will be traveling by train in Germany, Austria, and Italy, but will stay in Fuessen for three days. It has been suggested that I rent a car while in Fuessen, since that is really the only way to see everything in the area in 3 days. I am a bit nervous about driving in another country, so I am wondering if I really need a car, or will walking and buses suffice?
You don't need a car. I am sure whoever made that suggestion is not knowledgeable on transit in Germany.
There are regular buses all over the Füssen area. There is about one bus per hour between the Füssen Bahnhof and Hohenschwangau, site of the castles.
A lot of the buses to Hohenschwangau also go on to the Tegelbergbahn. Those buses also go on to Wieskirche, Oberammergau, and Garmisch-Partenkirchen. From Oberammergau, there is a bus to Linderhof.
Most of the old town of Füssen is a pedestrian only (Fußganger) zone. If your hotel is in that zone you would have to park outside and walk in, anyway.