Frankfurt is small enough that one careful day covers the headline sights, and a second day lets you slow down and eat properly. Here is the order I would do it in.
Römerberg and the Cathedral
The Römerberg is where most visits begin, and it is the square the city shows off: half-timbered houses, the city hall on one side, and festivals through the year. Walk it in the morning before the square fills up, then take the short walk to the cathedral.
Frankfurt Cathedral, St. Bartholomeus, is the Gothic tower that held the elections and coronations of the Holy Roman Emperors. The climb to the viewing platform is 328 steps, and the reward is the old town roofs on one side and the towers on the other.

The Main Tower and the Zeil
The Main Tower is the one to go up if you want to see how the two Frankfurts fit together. Its observation deck looks down on the medieval street plan and the bank towers at the same time, which is a view you do not get anywhere else in the city.
Back at street level, the Zeil is the long shopping street through the middle of town. The Goethe House, where the writer was born, sits a short walk from it and makes a good indoor stop when the weather turns.
The Museumsufer and the River Bank
The south bank of the Main is where the museums gather, and the Städel Museum and the German Film Museum are the two I would pick with limited time. If you plan on seeing more than one, the Museumsufer ticket is the better value.
Cross back over the Eiserner Steg footbridge for the view up the river, and walk the promenade in the late afternoon. In spring the boat tours on the Main start running again, which is a pleasant way to see the skyline from the water.
If you would rather not work out the order yourself, a Frankfurt tour covers this ground in a morning, cathedral climb and all.