As we know, the most common side effects from vaccines are mild. Below are two resources on side effects generally as well as an IOM report that looked at the evidence for all proposed side effects from immunizations, broken down by vaccine. This may be a helpful guide if people are concerned about a particular side effect from a particular vaccine (e.g. GBS from flu or encephalopathy from MMR).
One of the most difficult challenges we face is responding to the tragic stories of parents who believe they have lost their children as a result of vaccination. It requires a delicate balance of empathy and fact checking. Voices for Vaccines has created a blog post that outlines one potential approach: