Has anyone travelled from Fussen to Lindau? Is the train our only option?

Well, you can walk, bike, drive, swim or kayak the rivers (which would mean a huge detour, though)...

In case you mean to ask if there are any bus connections between these towns - no, there aren't. The trains do this job.

Look up train connections on http://www.bahn.de - however, remember that the town's name is not "Fussen" but Füssen, alternative spelling if you cannot type the ü is Fuessen with an extra "e". Otherwise you will get connections concerning a village named Fussenberg in the surroundings of Regensburg.

If driving the Bavarian Alpine Strasse or whatever it's name is skirts the foothills of the Bavarian Alps from Fuessen to Obertsdorf, a cutesy resort type place, and onto Lindau.

By train you change once to join the Munich-Lindau mainline - hourly trains at least on both segments i would think but use Quokka's bahn.de link for precise schedules.

If you rented a car for one day (usually no surcharges for one-way rentals) you would have at least two options for a really scenic day trip.
Hwys 310/308 (Bavarian Alpine Road) or
through Austria from Reutte on
Hwys 198/200/190 via Bregenz to Lindau.