The simple answer is that you need to take it one hole at a time.
I usually start in one of two places. 1. with the most complex/detailed part of the project, or 2. the part that is most important (like eyes/faces). Sometimes these two places are the same. If you are working on a project with eyes and the eyes get messed up, it will never look right, so I do those first (so that if i screw up I haven't wasted hours of cutting)
I always look the pattern over and identify any large peninsulas. Cut all the pieces on the inside of the peninsula before cutting the outside. I will often save the outside peninsula cut and tape it back into the hole to add support while I finish cutting the rest of the pattern. occasionally I will add extra bridges to a peninsula, but I don't do that often.
I also identify the areas of the pattern that are complex but don't require absolute accuracy. many times if you are not dead on the line, or if you connect a few holes, or add extra bridges or round over a few sharp points or whatever; no one will ever know. I remind myself often that this is supposed to be fun.
Lastly, or maybe it should be first, use a good quality baltic birch plywood or other fine grained hardwood.