frankfurt to strasbourg

I believe that the Lufthansa bus is slightly faster than the train, although they might run less often so it depends on when you want to leave Frankfurt. The bus leaves FRA for Strasbourg about every three hours (ave), takes 2½ hours, and costs €49 per adult. On the train, if you use regional trains, it takes about 4 hours and cost €38,50. By express, it's 2-3 hours, runs hourly, and cost €52.40.

cheapest and fastest don't always travel together. If planned ahead cheapest would probably get on the bus or train. Fastest would probably fly; fastest fastest would probably hire a private jet; neither would allow cheapest anywhere near.