A car would make it easier to include more options, but there's still nice places you can visit without a car, as mquartz pointed out, with Bad Homburg vor der Hohe and the popular tourist destinations along the Rhein. What time of the year is your trip?
We really enjoyed our days at Hotel Haus Lipmann in tiny Beilstein on the Mosel--but that would be 3 hours by train and bus from Frankfurt airport, versus 1-1/2 to 1/3/4 hours by car. And after you got there, you'd be dependent on bus schedules along the river to sightsee.
So, all in all, without a car stick with places not far from Frankfurt for that first day. It has been a number of years since we were there, so it might have changed, but we liked Bad Wimpfen which would be about 1-1/2 to 1-3/4 hours south from FRA by train.
As you do have 10 days, one place we really like is Bamberg. Good train links; lovely old town on the Regnitz River. Large enough to have everything you need and provide 2 or 3 days of sights and experiences, but small enough to be walkable and not feel overwhelmed. From Bamberg it's easy to get to places like Nurnberg, Coburg, Bayreuth. Lots of history--it's a UNESCO World Heritage City. And it is the beer epicenter of the German beer universe--highest per capita beer consumption in Germany and still has several smaller, independent breweries. One of its nicknames is "The town of pregnant men." Has the unusual distinction among towns north of the Alps of having both a Pope and an Emperor buried there.