Most side effects from vaccines are very mild. They're usually just some redness or swelling at the injection site. You can manage these by putting a cool washcloth on the skin to reduce the swelling.
Severe allergic reactions are very rare and are usually because of an allergy to Gelatin or Eggs (less then 1% of population), which are common ingredients in vaccines. Watch your child closely after they recieve their first shots and look for these symptoms:
Low blood pressure
Increased heart rate
If they have any of these things call your doctor right away. They could be having a reaction even if it's been a few days since the shot.