Frankfurt or Munich

I think that FRA, being a lot larger, is more forbidding than MUC. If getting out of an airport is easier than getting in, it might be easier to arrive in FRA and depart MUC.

If you're looking at direct flights to either MUC or FRA, it ought to be equal, but if you have to change at FRA on your way to/from MUC, then you have two things to consider. One, changing airports after a long trans-atlantic flight, when you are jetlagged is going to be more difficult than at the beginning of your trip, when you are fresher - advantage FRA first. On the other hand, if you are delayed getting into Frankfurt, there are probably more flights to Munich than there are to the U.S. - advantage MUC first.

Not sure which is better to fly in and out of, but Munich in my mind is a much more interesting city than Frankfurt. Or Cologne for that matter. Of course, this is based on 30-year-old memories!

In 2001, I flew in and out of Frankfut and in 2008, I flew in and out of Munich. They are both fine and I don't think it matters too much how you do it. Munich is quite a bit smaller and, as such, may be a little more user friendly. I would probably fly into Munich and out of Frankfurt-but really not a huge difference either way.

Where do you want to go first?
In terms of connections to the USA, Lufthansa offers direct flights to Frankfurt (its European hub) from Denver, and more limited connections to Munich from other US cities. Coming back, Lufthansa code shares with United, so flying back from London on a United flight makes economic and schedule sense. Frankfurt Airport and Hauptbahnhof are co-located, and are on the main train line to...about everywhere. Munich airport is about 40 minutes outside of Munich, and you have to travel to Munich Hauptbahnhof to make your onward connections. If you flew into Frankfurt, you'd be in Munich by noon on the ICE. There are plenty of Munich-London flights on the various discount airlines, so I'd suggest DEN-FRA, visit Cologne and Frankfurt, take the train to Munich, grab a flight to London, and LON-DEN. Or the reverse. The transfer between air and rail at Frankfurt is easier than you might think, though it's generally pretty simple in either FRA or MUC, even in a jet lagged state.
Only disadvantage to this plan is that Lufthansa can be more expensive, but if you've never flown them, you're in for a treat. Fortunately, DEN-FRA is a major route for them and they're fairly competitive.

I also vote for hitting Frankfurt and Cologne first and finishing up in Munich, as Munich is far more interesting than any of those cities. A much better coda to the German part of your vacation.

Thanks so much!! This helps a lot.