The book Twenty-Four Hours of Local Cohomology seems to be exactly what you are looking for.
It is an outgrowth by seven authors of a summer school on the subject held in Utah in 2005.
As the amusing title says, the book is divided into 24 short chapters, reflecting the 24 talks of the school.
These chapters are extremely easy and chock-full of examples and explicit computations.
The prerequisites are at a very low level and the first few chapters will probably be a just a reminder for you: definition of affine algebraic varieties over a field, sheaves and Čech cohomology, resolutions, limits, ...
Local cohomology is only introduced in chapter 7, in a friendly way.
A bonus of that book is that the chapters are so well constructed that one can learn a lot of independent material from them, even if one isn't interested in local cohomology!
For example Chapter 19 is a pleasant introduction to Grothendieck's algebraic version of De Rham cohomology; its relation to local cohomology is evoked only at the end of the chapter.
The level is so elementary that the first section starts with a proof of Green's theorem of "advanced" calculus!