It's the irony that you have to build up muscle strength in order to relax in order to use less muscle strength. In the meantime, if you work too hard, your tendons and ligaments take too much of that strain. What I always tell students is, if your fingertips hurt, you need to play more. If you have tension/muscular pain STOP. Stop right then. Do nothing else. Stop. Now the good news is that you don't have to stop for long. You can usually just shake out your hand and play after a few seconds. But you don't want that level of unnecessary tension to become habitual. I've often wondered if tendon strain becomes part of muscle memory; that is, your muscles train themselves not to do the work. At any rate, you need to develop strength and flexibility you don't yet have. It's easy to say that the ukulele is easy and lightweight, and easy to fret, but if your hands aren't used to some of the contortions, it's a LOT of effort.
I've played guitar for 20 years, and I've been working through some Hawaiian tunes which have BIG stretches in them on my concert neck (2107 and 2229 being the biggies - easyish on a soprano, not so on a longer neck). Those do make my fingers go ARG. The other was when I was in a punk band and I wasn't used to endless power chords. I had to work out ways of adding more relaxing hand positions to the tunes.
You have to allow your patience to win out over your enthusiasm for a while. Good luck!