Which side of the Rhein?

We've driven some, but not all of this route. You could leave the autobahn around Mainz or Weisbaden and make your way over to highway 9 on the west bank. Highway 42 runs along the east side. Drive as far as just south of Koblenz, and make your way back to the A3. You would pass through Bingen, Bacharach , Oberwesel, and past St Goar and Boppard. You can look at it on Google maps. Lots of castles on hilltops along the way. These are slow drives as you are going through towns and along secondary roads.

The most scenic stretch of the Rhine River begins just south of Bacharach and ends halfway between Saint Goar and Koblenz. You want the road that doesn’t detour away from the river between these two areas.

First a small comment to help you avoid confusion, the sides of the river are at least in Germany usually referred to as the right side and the left side. I really don't have an answer to you question, but the river looks pretty much the same no matter which side you are on. But if you choose the left side there is less risk of oncoming traffic blocking the view.

Something I forgot to add earlier- one of the reasons for my suggested route is that there are no bridges across the Rhine from Weisbaden/Mainz, until Koblenz. There are a couple of car ferries that go back and forth between the left and right sides, but no bridges. At least until they build the one at St Goar.

It's kind of a shame you aren't able to spend one of the two nights you refer to on the river instead in those larger cities. The small river towns often have lovely views of the river valley.

Drive the left side from Bingen north through Bacharach, Oberwesel, St. Goar and Boppard, which make for the best stops, all things considered. After Boppard, catch the Boppard ferry across the river and drive north through Filsen and Osterspai to Braubach; besides its attractive old-town center, Braubach is home to the best medieval castle experience in the area, Marksburg Castle, with tours in English at 4 pm.

If you find a way to spend a night, Bingen is a good choice, and the NH Bingen hotel is right on the waterfront.

I would take the left (west) bank. There are some nice towns to see like Bacharach and St Goar as well as being able to see the Lorelei from across the river as opposed to driving right under it. Depending on how you schedule things, you could take a brief side trip (about half an hour) from Koblenz down the Mosel to see Berg Eltz which is one of the truly worthwhile castles to visit. The road conditions are good and the driving has been pretty easy when we have done it, usually in the fall.

park the car and take a boat ride one way and a train back to the car. You see the scenery without a windscreen in the way and with no watching for traffic. Much easier for the driver. Much more scenic.

Watching the ships in the river is also great fun - with the telegraph indicators you can even predict what is around the corner. Best done from the deck of a boat, or from a deckchair on the porch of a hotel or in the riverside park at Bacharach, eating some quite tasty (don't tell JC) Schnellimbiss.

Nigel,
Okay, you made me look up what "Schnellimbiss" meant (junk food).

Will take the other side.
Going down the right side, you can visit Ingelsheim, Eltville as well as Eberbach Monastery, Assmannshausen, Rüdesheim (drive up to Germania overlooking the Rhein or take the chairlift), Lorch, go to the top of the Lorelei and look out, Braubach and the Marksburg. From the right, you can see all the castles on the left side, like Reichenstein, Rheinstein, Burg Rheinfels, etc. If you like you can take the car ferry across at St. Goarshausen and visit St. Goar, Burg Rheinfels, and Boppard.

Schnellimbiss is fast food. It doesn’t have to be junk food.

I prefer the west side

my map apps say on the highway it's about a 1.5 hour drive Frankfurt to Cologne

Hopefully, you can return the map app and get your money back.

The fastest Intercity Trains (ICE) go non-stop on a direct overland, non-senic route from FRA to Cologne in about ¾ hour, but they can go far faster than you can go in a car on the autobahn. ViaMichelin shows the fastest Frankfurt Flug to Cologne car route (non-scenic) at 180km (112 mi). There are 20 and 30 mph speed limits on some of the roads getting to the autobahn and a warning at one point on the autobahn of a 62mph speed limit enforced by camera. ViaMichelin estimates the time on this route at 1h53m.

And that is on a more or less straight line cross country route, not along the scenic Rhein gorge. The Rhein gorge route, according to ViaMichelin, is 210km (130 mi), with lots of 32mph sections, a takes over 4 hours to go from FRA to Cologne through the Rhein gorge.

which bank of the Rhein should we travel on, the East Bank or the West Bank?'

Actually, neither. If you travel on one bank, then you can see all the towns, castles, and other sites on the opposite side of the river, and you can visit towns on your side. For that objective, I think the left banks (as the river flows) would be preferable, because you get to stop in Bacharach and St Goar, which are, IMO, the more interesting places to stop, and you get to see the Lorelei from across the river.

But, in reality, the best route would be in the middle of the river, like on a KD cruise ship. That way you can see the castles and towns on both sides of the river, and you can see the middle-of-the-river toll castle, Pfalzgrafenstein, up close. You can't do that from either bank. If you take the ship from Bingen Stadt Bahnhof to St Goar, you can see the most interesting part of the river from the middle and stop and see Bacharach, in addition to your end point, St Goar.

The problem with doing it that way with a car, is that 1) you will miss a lot by having to drive, and 2) you'll have to backtrack by train from St Goar to Bingen to pick up you car. If this were my trip, I'd forego picking up the car at the airport, jump right on the train at the airport Regionalbhf and go to Bingen Stadtbhf (~1 hr), take the 1¾ hr boat trip to St Goar with maybe an hour stopover in Bacharach, end in St Goar and see that town (and maybe the castle), and then proceed on by train to Cologne. If you do all of this in one day, a 49€ Quer-durchs-Land-Ticket (for two) will cover all of your train travel plus get you a 20% discount on the cost of the KD tickets.

If you need a car from Cologne on, pick it up there.

Schnellimbiss is not junk food, (lousy translation, )standard fare when you want a Wurst...all different sorts.

Greetings,
My husband and I are working on a 2 month trip for the spring of 2021. One of our plans was to fly into Frankfurt and then rent a car there. I did some additional digging and learned that the train from the airport (runs almost hourly) takes less than an hour to reach Koln. Why take the train?
1. The train is fast and less expensive than the car rental and gas for one day. It leaves from the lowest level of the airport terminal. You will be in Koln faster than going to pick up the car and then the drive. There are several places that sell food to go, even a Starbuck, where you can pick up a meal or snacks to eat on the train.
2. There was and I believe still is significant major construction on the bridges that cross the Rhine around Wiesbaden.
3. When you are ready to leave Koln you can easily take the tram to the Koln airport to rent the car.
4. After the flight, you will not have the additional drive and depending on the time of day, having to contend with commuter traffic around Frankfort and Koln.
5. The train station is right next to the cathedral. Rick list many hotels in his travel book and you can find additional by using Goggle maps.
6. Use the money you save on the car rental and gas to take a wonderful cruise on the Rhein.
Hope this helps with your plans
Anne

Castleme2 writes,

When you are ready to leave Koln you can easily take the tram to the Koln airport to rent the car.

Or you can pick up a car at Köln Hbf - Köln's central train station.

Use the money you save on the car rental and gas to take a wonderful cruise on the Rhein.

A worthwhile cruise indeed, and it doesn't take long - about 1.5 hours to cruise from Bingen to St. Goar, the best part. And it's generally not a big hassle to sandwich a cruise into your Frankfurt > Cologne train trip. However, you must use a different train route from the one that takes just one hour (which does NOT involve the scenic part of the Rhine.) The Frankfurt > Bingen leg + the St. Goar > Köln leg will definitely require more time; this route takes you along the Rhine via the larger towns of Mainz, Koblenz and Bonn. You can see the fast route to Cologne on this map - and the Rhine River route as well:

https://upload.wikimedia.org/wikipedia/commons/c/cb/Schnellfahrstrecke_K%C3%B6ln%E2%80%93RheinMain.png

Or you could do a cruise of that same scenic Bingen > St. Goar segment from Cologne... first, catch a train to Bingen, cruise to St. Goar, train back to Cologne. Using a car from Cologne to Bingen would mean a "backtrack" train trip (or cruise) to Bingen to pick up your car after the cruise.