One of the things that you'll find as you learn more music, and figure out your chord shapes is the concept of keeping fingers where they are when you can to transition between one chord and another, or the typical shapes of chords (i.e. which fingers go where) that can be applied whenever you see a chord with a similar shape. Generally but not exclusively, it's "one finger per fret" kind of thing, i.e. forefinger on first fret, middle finger on second fret, ring finger on third fret, pinkie on fourth fret. This definitely is not definitive, but it's a good starting point. Again, if you're playing with chords, there are typical fingerings for common chord shapes, so start there, but it's also important to be able to use the same shape with different fingers, because of transitions in music that would make a different set of fingers make more sense in that particular moment.
I've shared this video before, but it's a really, really important set of exercises to build finger independence, which you will find helps enormously as you improve your ukulele skills, and build more songs and chords into your repertoire:
You may also find this video helpful:
One point from this video: you'll notice how she's bringing her pinkie finger in towards the fretboard by rotating from the shoulder, moving the elbow, not by rotating just at the wrist.
Good luck! There's so much to learn at the beginning, it just feels so overwhelming sometimes, but with consistent play, things start to become easier and make sense (I promise)!
Matt Stead, in his Beginner Ukulele YouTube series, talks about anticipating and planning for chord changes. If you know the sequence of the chords in a piece, then you can plan for alternative fingerings to ease the transition between chords. (Kind of like in band music, where you're often cautioned to "check the roadmap" before playing a piece to figure out things like repeats and alternative fingerings.)
I found this revelatory: You don't have to be bound by the proscribed fingerings of a chord. (Hope this touches on your question at least a bit.)
As for what fingers go on which frets when playing specific notes up and down the fretboard, my (beginner!) understanding is that you use index finger on first fret, middle finger on second fret, ring finger on third fret, and pinkie on fourth fret, and then go back to the index finger for the fifth fret, middle finger for sixth fret, etc.
Others will chime in with more detailed information.
I think a certain level of flexibility (figuratively speaking) is needed, as efficiency can vary based on progressions as stated above. Being able to predict how best to move down the neck can come from knowing the chord changes and trying to move from one to the other with the least change in fingering as possible. In other words, practice, practice, practice.
Interestingly, I broke my ring finger on my fretting hand as a senior in high school (in my 40's now) and cannot reach the 1 and 2 strings accurately as the end knuckle no longer bends. I have learned to compensate for this with my pinky and some odd stretches that now feel natural. It makes it very difficult for me to teach people the basics as I've had to throw them out the window.
Everyone is different- what I find simple and intuitive often gets me yelled at by other players in my groups - I don't play the chords "as written", for example.
Most of the books show an A7 as 0100 - coming from a guitar background, that chord shape should be (to me) 2130.
And when I'm doing the C-Am-F-G7 progression, I use anchor points - the shapes are (C) 0003 (Am) 2003 (F) 2013 (G7) 0212; I plant the ring finger and move it once in those 4 chords, plant the middle finger for 2nd & 3rd chords, only moving for the 4th chord, index finger plants and stays for 3rd and 4th chords.
That's 1st position - 2nd position barre chords have three shapes - 5433, 3453, 5553, & 7775 (last 2 chords are same shape, just slide up 2 frets).
Are they "by the book"? I dunno, I never read the book. In order to describe what I do, I had to play the chords and count the frets, then write 'em down, because I can't do tablature, either.
But if it sounds right, it is right.
Feel free to disagree.
And I can't address waxing the neck, as Nickie suggested, but if you are playing outside in the Northeast in the summer, bring some cornstarch-based unscented powder, and a towel. You will need 'em.
I am not sure how helpful this will be but one of the best ways I found to practice when I was new to the uke was through the 'worm'. I think it really helps build up one's muscle memory and to get used to where the fingers should 'normally' be.
Phil Dolman explains it as the first exercise in this video below: