For my thinking will be nice to have current 5A from 1,25V to 24V.
You need good transformer.
You can use linear LM338K (my favorite) or switcher regulator LM2678T-ADJ both are 5A. Price for both in my country is about 8eur. Circuits are simple and you can find it easely on Internet.
For measuring and showing actual values on LCD you can see this link :
That circuit from tpetar (the one with the op-amp) does exactly what you want, and has LED indication too. It looks like a good circuit. If there was a simpler circuit, it would not be called a circuit, it would be called a "ready-made power supply" : )
Also, if the -5V is such a concern, you could just connect to 0V, the -5V means the supply can go all the way to 0V.
I've not tried it, but that's what it looks like.
But you say you want 3-5V output voltage up to 1,5A. You dont need nothing special for that.
Todays universal lab power supply should have small ripple voltage, with at leat several ampere capability on output, and voltage range from 1V to 24V, this is my thinking.
variable voltage from 3v to 5v, any value in between ex 3.2 3.7 4.0 4.5 etc.
current 0 to 1.5 amps ; I can set any value in between.
Led indicator in case of current over limit