Booked under Lufthansa, have MP and Amex points. Help?

Hi,
I'm new to using points and I'm confused.
I booked a flight on Lufthansa, operated by various carriers (including some legs by United). I have 60k MileagePlus points and 150k Amex Platinum MRF points.
I'm just not sure what I should do to upgrade the flights. Transfer from Amex to a Star Alliance partner, then transfer to Lufthansa for the upgrades? The United site also allows me to upgrade their legs, but they aren't the booking carrier, which makes me nervous.

Just not sure how to get started navigating all of the partnership complexity...

Thanks for any guidance!

Ryan