There's not a kids mode per se on the iPad mini. But all iOS devices have parental restrictions you can enable to restrict certain apps, installing apps, deleting them, reporting your location etc.
The original mini (released two years ago) is $249 new, and last year's model is $299 new. But that's from Apple. This time of year you can find them cheaper from Walmart, Target, etc. And for the refurb route, Apple certified refurbs come with a full 1 year warranty. If money isn't an issue, I'd recommend going with the iPad mini 2.
My youngest was five when the iPad minis came out and we put it in a case like this (but pink):
View attachment 73205 which she still loves. She'd been using my full size iPad since she was two.
To add a bit more, it is great for long car trips because you can download movies on it. There are lots of educational apps with basic stuff like learning the alphabet and counting. Disney, Nickelodeon, etc all have apps with kids' favorite characters. Some are just games, some are more educational. There are also lots of kids' books like Dr Seuss where the book is interactive with things to touch on the page and you also have a choice to have it read the book to your kid (for when you are tired of reading the same story for the hundredth time) or turn off the voice and read it yourself.