What kind of examples are you looking for? I mean there are plenty of examples, just take an explicit singularity and start blowing up. What properties are you looking for? If you are looking for behavior that does not happen in low dimensions, then an interesting family of examples come from what is known as a small resolution, that is, a resolution where the exceptional set is not a divisor.
I believe the simplest example of a small resolution is given by a cone over $\mathbb P^1\times \mathbb P^1$. The resolution is given by blowing up the surface that is the cone over one of the ruling curves on $\mathbb P^1\times \mathbb P^1$. Since the (big) cone is smooth away from the vertex, this blow up will not do anything there and over the vertex it will have an exceptional curve which is isomorphic to $\mathbb P^1$ and really corresponds to points on an(ya(ny) ruling curve transversal to the one that gave the blown up surface. This has all the nice properties you asked for: the singular set and the exceptional set are both smooth, it's even an isolated singularity. And, of course, you couyld could have blown up the singular point and get the entire $\mathbb P^1\times \mathbb P^1$ as the exceptional divisor.
A similar example can be cooked up from resolving cones over products in general.
Perhaps the next example to consider is when the singular set is larger dimensional, but not simply because it is (say) a product of an isolated singularity with something else.
Yet another way to find many interesting examples is by quotients.
Or you could just try to take your favorite projective variety and project it to a smaller dimensional subspace and then try to resolve the singularity. Although this can get really messy really soon so you have to choose the starting variety carefully. (Dolgachev has a paper on general surjection surfaces or something like that with generalizations by Steenbrink and Doherty (two papers).)