Peeling citrus strips

I'm quite good in the in kitchen and have good knife skills but I cannot for the life of me figure out how to peel nice zest strips like I see in videos or when I go out. I've got the right peeler, I get fresh fruit, but I always manage to peel thin strips that don't have structural strength or I end up cutting myself (like I did last night...) when the peeler slips off the fruit. I've watched videos and I just cannot figure this out.

Does anyone have a good recommendation for techniques to peel zest strips so I can enjoy my damn drink instead of bleeding all over the place first?!