Dec 4-10th, Stay in Cologne or Frankfurt?

You seem interested in the more popular Christmas markets. The Stuttgart and Esslingen markets are highly praised on this forum and elsewhere. I'm a bit surprised that you are looking at Frankfurt and Cologne instead. If you stayed in Esslingen, you'd have more "medieval old town atmosphere" than you can shake a Lebkuchen at. And getting to FRA for your final night from either town is pretty much a snap.

https://community.ricksteves.com/travel-forum/germany/can-i-see-enough-of-cologne

If you are in Frankfurt during that time period, you could visit some of my favorite markets. Especially Büdingen, Idstein, Seligenstadt and Marburg. All of them are within 1 hour by regional train from Frankfurt. If you are in Frankfurt on a Sat. night, you will want to hear the brass band playing from the gallery on the Alte Nikolai which looks over the old town.
Unless you have a super early morning flight out of Frankfurt, there is no reason to stay at the airport. Trains run every 15 min. from the city center and take about 13 min. to get to the airport. Central hotels would be Motel One, Westin, Moxy, Flemings City Center, Hilton or Steigenberger.

Michelstadt Advents Market 24.11-23.12
Wiesbaden Sternschnuppen Market 25.11-23.12
Speyer Christmas Market 25.11-6.01
Seligenstadt Christmas Market 27.11-14.12
Mainz Christmas Market 27.11-23.12
Limburg Christmas Market 28.11-22.12
Marburg Advents Market 28.11-23.12
Idstein Advents Market 5-7.12
Königstein Christmas Market 5-7.12
Büdingen Christmas Market 3-7.12d

Diana, maybe you already found it, but here is a list of the markets in Budapest. https://community.ricksteves.com/travel-forum/hungary/budapest-christmas-markets-d26f8ab1-2048-4160-bfb3-bf13b096683e

Sounds like a wonderful trip. Make sure you do a trip report.

Ms. Jo is correct; you would do better to stay in Frankfurt than Cologne. My reasoning though, would be slightly different. Frankfurt is much more viable if you want things to do and see outside of the markets. And, while the markets are best come sundown, cities like Mainz, Heidelberg, and Speyer. all of which are very easy to reach by train from FRA, offer a huge variety of "museums, old town atmosphere, and good eats".

Mainz has the Gutenberg, and all the Roman ruins, including the Temple of Isis, along with some really good places to eat. Speyer has the Cathedral and Technical Museum. Heidelberg has the castle and an impressive shopping area in the center of town. And all of these have very good Christmas markets.

But from FRA you also have the opportunity to visit some of the smaller, really interesting, places. On top of those offered by Ms. Jo I'll suggest Bad Kreuznach, Bad Durkheim, Neustadt (Winestrasse), Deidesheim, and Darmstatt (which is a big city, but has excellent museums for history and art). All of these are also easy trip via the regional trains.

But don't cut Frankfurt short. There's a lot to do right there.

I know so little about Germany that I was going to try and stay out of this one. I have flown into Frankfurt a half dozen times, but only stayed long enough to catch my ride to where my son lives about 2 hours away. So don’t know a thing about Frankfurt, except a nice airport and the city doesn’t seem far away so I probably wouldn’t lose a night in the city to stay near the airport. But that just uneducated me.

I have spent time in Cologne. About a week in and around when my daughter was in university nearby. I was not impressed. No desire to return. But again, just me.

So, based on my ignorance I chose Frankfurt. BUT, I chose Frankfurt for another reason as well; that would be Ms. Jo. Nothing improves the quality of a holiday than good reliable first hand input from someone with a verifiable resume (look at here profile page). I post your exact dates and Itinerary and then if she is in the mood, she might be able to give you some detailed assistance. Its easy to throw out a big wish list of recommendation but its useless without context and she can maybe help with that.

I chose Frankfurt for another reason as well; that would be Ms. Jo.
Nothing improves the quality of a holiday than good reliable first
hand input...

Ms. Jo lives in Frankfurt and operates walking tours there. Unlike most of us without the option of hitting German Christmas markets every year, Ms. Jo hits them regularly and posts frequently about these visits. Besides my own impressions of Esslingen, it is Ms Jo's big thumbs up for both Stuttgart and Esslingen that mostly prompted my suggestion for these places.. I am linking below to some of her previous comments... You'll read comments from others as well by clicking on the links.

You might want to get to Esslingen a bit earlier as the lights and
torches start going on at dusk, which will be around 15:30-16:00. This
gives you a bit of time to see Stuttgart at night too, as it the
market is really pretty. All the stands are decorated. It is not like
Nuremburg, where every stand looks alike. The roofs especially are
cool... For those who haven't gone to Stuttgart in many years,
consider giving it a try again. The city has more to see and discover
than you might think. Palaces, vineyards, and history.

https://community.ricksteves.com/travel-forum/germany/esslingen-or-stuttgart

"I too really enjoyed the Stuttgart and Esslingen markets. They are 2 of my favorites and I have been to a LOT of markets. Would like to see Rick devote a bit of time visiting these 2 cities as they deserve more attention."

https://community.ricksteves.com/travel-forum/trip-reports/trip-report-christmas-markets-in-and-around-stuttgart

THANK YOU ALL for the helpful replies! So---Frankfurt it will be. Now to find a hotel, or preferably an Apart-Hotel.

Diana-
Hello. We are planning a trip about the same time to do some Christmas markets. We have friends that live in the vicinity of Marburg (north of Frankfurt). We mentioned to them Frankfurt or Cologne markets, but they didn't really want to go to them because of the crowds. They look awesome, but are very crowded I guess. Per Ms. Jo we are going to hit the Marburg, Idstein and Budingen markets for the smaller town feel.
No one has really addressed your other possible day trip from Nuremberg. I have been to both the Nuremberg and Regensberg markets- both are very nice. I particularly enjoyed Regensberg! You might want to consider Bamberg or Bayreuth for your other day trip. I've heard that they are quaint and nice. I think we are going to try to get to Bamberg's market while we are there. We stayed in Bayreuth for Christmas pre Covid and enjoyed it immensely!

Best of Luck,
Dan