Easiest way to determine the singular locus of projective variety & resolution of singularities
The other answers are already very good. A few additional notes:
(1) As others have explained, the Jacobian ideal method works for projective space. It also works for other toric varieties. Any smooth toric variety can be written as $(\mathbb{C}^n \setminus \Sigma)/(\mathbb{C}^*)^k$ where $\Sigma$ is an arrangement of linear spaces and $(\mathbb{C}^*)^k$ acts on $\mathbb{C}^n$ by some linear representation. For example, $\mathbb{P}^n = (\mathbb{C}^{n+1} \setminus \{ 0 \} )/\mathbb{C}^*$. So you can use Greg's trick of unquotienting, using the ordinary Jacobi criterion, and ignoring singularities on $\Sigma$. See David Cox's notes for how to write a toric variety in this manner.
(2) For varieties of dimension greater than $1$, resolution of singularities is very computationally intensive. It has been implemented in Macaulay 2, but it tends to tax the memory resources of the system.
The Jacobian condition for smoothness is valid also for projective varieties as well as affine varieties: you just take a homogeneous defining ideal and compute the rank of the Jacobian matrix at the point p, see e.g. p. 4 of
http://www.ma.utexas.edu/users/gfarkas/teaching/alggeom/march4.pdf
For a general variety: yes, I think the most computationally effective way to do it is to cover it by affine opens and apply the Jacobian condition on each one separately.
About your question on resolution of singularities: this is tricky in high dimension! As I understand it, you can indeed resolve singularities just by a combination of blowups and normalizations (at least in characteristic 0), but starting in dimension 3 you have to be somewhat clever about where in and what order you perform your blowups. It is not the case that if you just keep picking a closed subvariety and blowing it up (and then normalizing) that you will necessarily terminate with a smooth variety.
For more details presented in a user-friendly way, I recommend Herwig Hauser's article
http://homepage.univie.ac.at/herwig.hauser/Publications/The%20Hironaka%20Theorem%20on%20resolution%20of%20singularities/The%20Hironaka%20Theorem%20on%20resolution%20of%20singularities.pdf
Concerning your first question:
For many questions, the easiest way to see the nuts and bolts of a projective variety $V \subseteq P^n$ is to look at its cone $CV \subseteq A^{n+1}$. After all, the graded ring whose Proj is $V$ is the same as the ungraded ring whose Spec is $CV$. Obviously, there is almost always a singularity at the origin; but if you ignore that point, the other singular points all correspond between $V$ and $CV$. You can also think of the grading as geometrically represented by multiplication by $k^*$, if you are working over an algebraically closed field $k$. (Because the homogeneous polynomials are then eigenvectors of that group action.) You can think of $V$ as obtained from $CV$ by and then dividing by scalar multiplication.
The atlas-of-charts analysis of a projective variety is certainly important, but to some extent it is meant as an introduction to intrinsic algebraic geometry rather than as the best computational tool.
Your second question is reviewed in Wikipedia. As Wikipedia explains, Hironaka's big theorem was that it is possible to resolve all singularities of a variety by iterated blowups along subvarieties. I do not know a lot about this theory, but if so many capable mathematicians went to so much trouble to find a method, then surely there is no simple method.
On the other hand for curves, there is a stunning method that I learned about (or maybe relearned) just recently. Again according to Wikipedia, taking the integral closure of the coordinate ring of an affine curve, or the graded coordinate ring of a projective curve, solves everything. The claim is that it always removes the singularities of codimension 1, which are the only kind that a curve has.