Some new ukuleles (especially the Chinese ones) come with atrocious strings. Use a good brand and remember that any string will stretch like crazy for the first few days.
2. Sticky Nut
The grooves in the nut (the plastic thing at the top of the neck) may be grabbing the strings because it is rough or too tight. The solution is to gently pop the string out of it's groove and use a teeny little file and make the grooves smoother and wider if necessary. When you do this DO NOT make the groove any deeper, just smoother and wider. If you worry about that you can also just blacken the groove with a number two pencil as the graphite will act as a lubricant on the strings.
3. Cheep Tuners
Cheep tuners are a curse on all humanity. These can be replaced.
4. Bridge Out Of Place
The worst possibility is that the bridge is improperly placed. I don't recommend moving it. You should get a luthier to do it.
