Well, the transition metals (the bulk of the table in the center) are weird and don't follow any particular easy rules. They also have the issue of having variable charges and such due to that d-orbital, so really, the only way I've ever seen to know them is to just memorize the common charges for each element. Chromium (IV), Chromium (VI), Mercury (II), Iron (III), etc. It's a bit of a pain.
Through for the A-group elements (or Groups 1, 2, then 13 to 18 on your new-fangled Periodic Tables), you can tell charge based on what group they're in. Group 1 and 2 elements have +1 and +2 charges, then jumping across the trans-metals Boron's group has +3, Carbon's +4, and Nitrogen's +5. As suggested for the last two groups you can as a rule of thumb count back from the Noble Gases to guess the charge. The Halogens have -1 charges, the Chalcogens (Oxygen's group) are -2. Period Law is fun, but complicated at times.