It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about the modern AG. But is it possible for someone who is out of the Academia and has not much background, except a typical Undergraduate Alegebra and Some Analysis, to just go through the book, page by page? If not, what is the proper rout for entering into a serious Algebraic Geometry book, like Hartshorne's?
Hartshorne's book is an edulcorated version of Grothendieck and Dieudonné's EGA, which changed algebraic geometry forever.
EGA was so notoriously difficult that essentially nobody outside of Grothendieck's first circle (roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .
Things began to change with the appearance of Mumford's mimeographed notes in the 1960's, the celebrated Red Book, which allowed the man in the street (well, at least the streets near Harvard ) to be introduced to scheme theory.
Then, in 1977, Hartshorne's revolutionary textbook was published.
With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.
However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.
There are many such books nowadays but my favourite is probably Basic Algebraic Geometry, volume 1 by Shafarevich, a great Russian geometer.
Another suggestion is Milne's excellent lecture notes, which you can legally and freely download from the Internet.
The most elementary introduction to algebraic geometry is Miles Reid's aptly named Undergraduate Algebraic Geometry, of which you can read the first chapter here .
Miles Reid ends his book with a most interesting and opinionated postface on the recent history and sociology of algebraic geometry: it is extremely profound and funny at the same time, in the best tradition of English humour.
Detailed recommendations to start can be found in this analogue question or this other one (UPDATE: this recent long answer details a learning path from HS mathematics to advanced algebraic geometry), may be useful to you. I think the best route is firstly get a classical background on algebraic geometry (and any geometry in general) with the wonderful book by Beltrametti et al. - "Lectures on Curves, Surfaces and Projective Varieties", which serves as the most traditionally flavored detailed introduction for Hartshorne's chapter I, but done rigorously and with a modern style (proof of Riemann-Roch without mentioning cohomology!). This can be accompanied by the first half of the wonderful new title by Holme - "A Royal Road to Algebraic Geometry", where a beautiful treatment of curves and varieties is introduced in a very pedagogic way, and whose second half introduces the categorical formulation of schemes and sheaf cohomology and serves as an aperitif for Hartshorne's hardest chapters. Follow some of the rest of recomendations at the other answers, and patience and dedication will lead you to master Hartshorne. The main hard work you will need to do is a good mastery of commutative algebra; for a great thorough treatment of abstract algebra, I would recommend Rotman - "Advanced Modern Algebra" which is really nice for self-learning (but 1000 pages long!) and treats much of the background you need, to be supplemented with parts of Atiyah/MacDonald - Introduction to Commutative Algebra (you may start getting your algebraic background with Reid - "Undergraduate Commutative Algebra") or with the new Singh - "Basic Commutative Algebra" (Einsenbud's big book is a standard for many people as algebraic background for Hartshorne, but I see Atiyah/MacDonald+Sigh a more concise, more complete formal approach). For a full lists of references on classic algebraic geometry (curves and varieties) and modern (schemes and beyond), I have compiled two detailed lists at Amazon: Definitive Collection: Classical Algebraic Geometry and Definitive Collection: Abstract Algebraic Geometry.
Try an "Invitation to Algebraic Geometry" by Smith, Kahapaa, Kekalainen and Traves. (Springer).
In order to supplement Georges nice answer, I recommend you have a look at William Fulton's Algebraic Curves. It is a great book, which covers elements of the theory of algebraic curves from a "modern" point of view, i.e. with a view towards the modern approach to algebraic geometry via schemes (although this is never explicitly mentioned anywhere in the book).
I personally like it more than Reid's book, which Georges mentions. I think Fulton's is more systematic and thorough (at the risk of getting a little bit terse at times), while Reid's follows a more pictorial, scrapbook-stlye approach (at the risk of getting disorganized at times).
I ask the great algebraist Vinberg whose contemporary text on ALGEBRA (AMS pub) what parts of his book should be read so as to make Hartshorne's legible: he said the chapter 9. But this is rather a short chapter ! 6o PP.
As an aside his book is very similar to Kostrikin's. Student's Kostrikiin were told that is all you need to know, from which you can go deeper into the subject.
One feels with Kostrikin's that he is directly talking to the student, strange quality of some texts and some teachers. That one is directly getting it from the master.