Quillen's "Higher Algebraic K-Theory I" is probably the best source for understanding the basics and the original intuition.

Thomason/Trobaugh is also an excellent paper, but it is a fairly large paper and very fundamental (so the first half of the paper is dedicated to construction of the basic objects).

And if you want a deeper understanding, you could have a look at some of Thomason's older papers, as well as some of Waldhausen's papers.

When I was learning Algebraic K-Theory, I kind of found it easier to understand by going backwards (i.e. I would think of something to get a kind of big picture, ask myself questions about why something might be true, and use that approach to go backwards through Thomason/Trobaugh and if necessary back to older papers). Not everyone will agree with this approach, but I felt that it helped me to build the intuition needed to progress in the subject.