For static photos, I'd suggest A mode. The depth of field can be less than you think, depending on how close you get. Following example was taken at f/11 and just about gets the eyes and beak into focus.
If you're much further out then you can get away with bigger apertures. Do review at the time, and/or take shots at wider and stopped down to hedge the output.
Like Justin said, they do tend to move around, and the longer exposures will open up a motion blur risk, plus I'm sure some of their eyelids are synchronised to my shutter. Taking several shots will help you get a great one.
Oh, if the bird is quite contrasty like the harris hawk above, lower ISO will give you more dynamic range to keep the detail in both dark and light areas. When I first took the above shot I had auto ISO on, and I found I was borderline clipping the highlight detail (beak). Reducing to ISO400 solved that for me at the time.
For flying birds with unpredictable paths, I'd suggest S mode set to 1/500s or even faster if light allows. I would use continuous AF and continuous shooting, centre point AF. Keep the bird centred and crop to frame if needed when you get home. Expect a significant number of duff shots. If they fly repeatedly on a predicable flight path, you can try manual pre-focus instead and wait for them to go into the target zone.