What is the most intuitive dependent type theory I could learn?
There are a few different ways of approaching this:


Dependent Type Theory.
This is probably not the easiest way of learning about dependent types, but you could look at the Calculus of Constructions papers and their variants, Pure Type Systems, and Martin Hofmann's article on the Syntax and Semantics of Dependent Types, for example.
Theorem proving. Firstly, take a look at my answer to a related question: How would I go about learning the underlying theory of the Coq proof assistant?. 
Programming with dependent types. Looking at recent languages with dependent types  such as Epigram or Agda or less recent ones such as Dependent ML and writing some programs will help solidify the ideas. Epigram, for example, is extremely clean in design. Another angle is to see how dependent types are implemented. One practical dependent type theory is $\Pi\Sigma$: Dependent Types without the sugar. Several PhD theses are devoted to programming with dependent type theory: Dan Licata's, Nils Anders Danielsson's, Ulf Norrel's, Susmit Sarkar's, among others. And of course there's Adam Chlipala's book, given in Marc Hamann's answer.


I'd be inclined to start with programming, before moving to using theorem proving, then start exploring the more theoretical issues.