I've tried learning algebraic geometry several times. I asked around and was told to read Hartshorne. I started reading it several times and each time put it away. I realized that I could work through the sections and solve some of the problems, but I gained absolutely no intuition for reading Hartshorne. Discussing this with other people, I found that it was a common occurrence for students to read Hartshorne and afterwards have no idea how to do algebraic geometry. (I imagine this was the motivation for asking this question.)
After more poking around, I discovered Mumford's "Red book of Varieties and Schemes". While Mumford doesn't do cohomology, he motivates the definitions of schemes and and many of there basic properties while providing the reader with geometric intuition. This book isn't easy to read and you have to work out a lot, but the rewards are great. Another great feature of this book is that Mumford bought the rights to the book back from Springer and the book is available for free online.
Another book was supposed to be written that built on the "Red book" including cohomology. After many years, I think this is near completion; see Algebraic Geometry 2. Whlile many of the above books are excellent, it's a surprise that these books aren't the standard.