Any advice for how best to add French Horn to my arsenal given that I already play trumpet and flugelhorn? I'm quite proficient at the two and want to add a third instrument this year.
It's very different! The mouthpiece is smaller than a trumpet's, yet the instrument's range covers that of a trombone. And, while you can get away with playing flugelhorn like a trumpet, if you play the horn like a trumpet not many people are going to want to listen to you.
My advice would be to essentially relearn the horn fingerings from scratch instead of trying to transfer them from your existing valve chops (not to mention they're on a different hand).
The middle range of the french horn is two whole partials above where it sits for other brass instruments, and that means all of the partials for much of your playing are going to be MUCH closer together. It will be great training to have to really zero in on the partial you're trying to play more than you would have to if you were playing trumpet.
Other things to watch out for in basic french horn technique:
You will need to train your brain to go from "french horn mode" to "trumpet mode" and back. Skills will transfer from one to the other, but you should keep them distinct and integrate technique into them separately.
If you are playing a double horn, you may realize that the Bb side can be played just like a trumpet. If you're just goofing around, then fine, but DON'T do this as a way to learn french horn. The names of the notes are different since transpositions are still for F horn, and you need to learn the appropriate spots to switch from one side of the instrument to the other while staying in "french horn mode".
Adding French Horn to Trumpet and Flugelhorn? My advice woud be absolutely "yes". But try to think as a french horn player. Requires practice to change the mode. That's a great exercise for your brain. Have fun!!!