Two mics
Using two mics at the same time, you'll want to avoid phase problems caused by sound waves from each source hitting the "wrong" mic at a slight delay. That means either going with a stereo setup, which isn't what you asked for, or keeping the mics relatively close to their respective sources, so the bleed from the further source is significantly lower in volume compared to the primary source.
Obviously, there is a natural position for the vocal mic close to your mouth, so do that. ;-)
As for the uke, there are no hard and fast rules. A lot depends on the uke, the mic, and your playing style. My ideal would be to pull the mic back a foot or two to get the "whole instrument," but the further away you get, the more the uke mic will grab your vocals...especially if you're a loud singer. I recommend doing some test recordings and seeing what works best for your setup. DON'T judge by what you hear on computer speakers. If you don't have access to a good sound system, burn your test versions to a CD and listen to them on a variety of systems: your car, headphones, next door, etc.
Make a series of recordings with the mic aimed at the sound hole, at the edge of the sound hole, the body, the "sweet spot" where you strum, etc. "Slate" your recordings with the mic position so you know what works better and what doesn't. Don't be afraid to play around. Recording is both a science and an art.
As for stands, I like "one knob" booms mounted on tripod stands. Ultimate makes nice stands with a pistol grip--you squeeze it and adjust the height. Very handy and saves a lot of time. Use tripod stands. Weighted base stands couple more strongly with the floor and are more likely to pick up unwanted vibrations.
What kind of mics are you using?