Well, you'll really want to read them all at some point. To start with, take Griffiths-Harris for geometric insight and Huybrecths for company (his chapter 1.2 is amazing). Voisin is very good and at first covers the same ground as Huybrecths, but is more advanced (do read the introduction to Voisin's book early, it sets the scene quite well). Demailly's book is where all the details are, you'll want that one for proofs of the main theorems like Hodge decomposition, Kodaira vanishing etc. There's also a new book by Arapura that looks very user-friendly.
And now for some clearly false generalities: The books by Huybrechts, Voisin and Arapura have very algebraic points of view; they were written by people who are mainly algebraic geometers and (to simplify greatly) think in Spec of rings. By contrast, Demailly and Griffiths-Harris have more differential-geometric points of view and use metrics and positivity of curvature as their main tools. I'll take the opportunity to also recommend Zheng's wonderful "Complex differential geometry" for an alternative introduction to that point of view. You'll need to know how to use all of these tools (as do all those people, of course).
So, to sum things up:
$$
\begin{array}{ccc}
& \hbox{introduction} & \hbox{advanced} \\
\hbox{algebraic} & \hbox{Arapura, Huybrechts} & \hbox{Voisin}\\
\hbox{metric} & \hbox{Griffiths-Harris, Zheng} & \hbox{Demailly} \\
\end{array}
$$
The best complete, concise, clear and rapid overview of the essentials of Riemannian geometry that I know is found in the little book Morse Theory, by John Milnor. As far as delivering a more detailed review of this book: well, I've already said Milnor; need I say more?
Note Added Wednesday 8 July 2015 9:33 PM PST: Despite my previous invocation of the honored name Milnor as a more or less complete justification of the utility and quality of his Morse Theory as a most excellent source for quickly grasping the basic facts of Riemannian geometry, I felt it might be helpful to add a little bit about the contents of this work, if for no other reason than to fully answer our OP's question, and thus if possible save her or him some time in finding the right place to learn what he/she needs to know. The book, as the title indicates, is directed towards Marston Morse's theory of critical points, an essential tool of differential topologists; but it focuses in the large on the Morse theory of geodesics, applying his critical point theory to the relationship between convergence/divergence of geodesics and manifold topology. In this book, as elsewhere, the critical-point analysis is used to discover facts about the the topology of path spaces, and ultimately to prove the Bott periodicity theorems to reveal aspects of the structure of the higher homotopy groups of the Lie groups $O(n)$ and $U(n)$ etc. This is done via the study of conjugate points, i.e., points along a given geodesic where Jacobi fields, that is, those vector fields $W(t)$ along the geodesic $\gamma(t)$ satisfying
$\nabla_{\dot \gamma} \nabla_{\dot \gamma} W + R(\dot \gamma, W)\dot \gamma = 0, \tag{1}$
where $R$ denotes the Riemann tensor, vanish, having been initialized with $W(0) = 0$, $\nabla_{\dot \gamma}W$ arbitrary.
Now I know very little about algebraic geometry; indeed, I am learning someting of it, slowly and surely, mainly by reading posts here and on Math Overflow. But I suspect that the main utility of Riemann's geometry to the algebraic kind lies in the theory of characteristic classes, which quantify certain important properties of tangent and other vector bundles; here the Riemann tensor field plays an essential role, e.g. the Euler cohomology class, whose evaluation on the top homology class $[M]$ of the manifold $M$ under study, yields the Euler-Poincare characteristic number, may be expressed locally as a polynomial in the components of $R$; the Gauss-Bonnet theorem is the perhaps the most elementary example of this result. So perhaps Milnor's book goes in a slightly different direction than would best suit the needs of an algebraic geometer; but in terms of learning the basics, it is, in my humble opinion, hard to beat. And after reading Morse Theory, one can always tackle Milnor's Characteristic Classes, which may indeed be more suited to the needs of those pursuing algebraic geometry. End of Note.
Best Answer
I think the best option with little knowledge of differential geometry is definitely Huybrechts' Complex Geometry. His approach is to introduce just the right amount of differential geometry that one needs to understand the proofs of the big theorems. Plus, he mixes in the sheaf theory with the analytic theory in a nice way. It will definitely serve you well if you know at least a bit of de Rham cohomology before starting. If you go with this, don't be afraid to skip chapter 1.2 on first reading and refer back to it when you need it. That chapter has a fantastic treatment of the linear algebra needed for complex geometry, but you might feel more motivated to read it once you know how it will be used.
Voisin's book is a better second course in my opinion. This set of books goes much deeper into both the analysis and the homological algebra of the theory. I've found the proofs in that book to be of greater depth than Huybrechts'. Plus, no analysis punches are pulled, as opposed to Huybrechts' book. So, if you feel like you want to read the proof of the Hodge decomposition, Voisin's book has the detail you'll need.
Griffiths and Harris and Demailly's books are excellent from the analytic viewpoint, but without differential geometry background I would wait on these. They are both almost entirely differential geometric in nature. G&H's chapter 0 does have a ton of great background material for complex geometry that is nice to refer to on occasion, though.
Demailly's book is hard. It is great in that it really starts at the beginning with definitions of smooth manifolds. That being said, if heavy analysis isn't your cup of tea, then that book will be a tough read. I use it more as a reference when I need to dig deep into analytic details.
CMP's book is rather orthogonal to the rest on this list. If you're just trying to learn algebraic geometry, I would suggest reading the first chapter up until Hodge structures are introduced. The way they illustrate the case of elliptic curves and tori is a really nice intro to period domains/maps, but I personally found the next few chapters incomprehensible on first pass. I personally felt that the details were far too sparse in this book.