"I don't fully understand how to resize my UIView containers to resize themselves to fit smaller screens." -UIViews won't resize according to the screen, you will have to add constraint for them so that they pick their width and height. Alternatively you have to set constraint for fixed height and width.
In my opinion, for supporting smaller screen size, you must use scrollviews as parentview. So that user can scroll in the app. Also you can give relative width and height for views inside scrollview.
Scrollviews are required because you will always want some minimum width and height for buttons, labels etc. Otherwise on larger screen like Ipad they will be very large, while on smaller screens they will be very small.
You can use TPKeyboardAvoidingScrollView: How to use TPKeyboardAvoidingScrollView, or just google for that. It handles keyboard showing and hiding task, which is a headache otherwise.
Here is how I think you can solve your issue:
Set Some minimum height and width for your topmost view(scrollview preferably).
Now add other views inside it and use relative width and height. In relative width and height you give values in ratios. Here you can get an idea how to do that: -Giving width in % values in autolayout.
Also add additional constraint on your internal views for minimum height and width so they don't fall below certain size.
I would not suggest, but you may always use UIScreen.main.bounds.size.width
and UIScreen.main.bounds.size.height
to get the width and height of screen. And according set values for your constraints in your view/controller class. Here is a link for setting value of constraint in swift class:
set contraint value programatically