Childhood Ailments and Diseases
I have divided this section into two parts. The first is mainly harmless but bothersome ailments, which will affect almost all children at some point. The second is diseases, some of which are common and some which aren't, but all are on the CDC's recommended vaccine schedule as illnesses we should vaccinate against.
*Standard side effects applicable to all vaccines are: pain, redness, swelling at the injection site, fever, crying, vomiting, diarrhea, poor appetite, sleepiness, headaches, body aches, pea-sized nodule at injection site lasting several weeks, whole body rash or area rash
Explanation of several more severe side effects:
Guillain-Barre syndrome: body's immune system attacks the nervous system. Causes temporary (several weeks) paralysis and weakness. Sometimes fatal.
Encephalitis/encephalopathy: swelling and inflammation of the brain. Baby will have a high fever and scream intensely for several hours or days, alternating with lethargy. A small percentage will experience permanent brain damage.
Subacute sclerosing panencephalitis (SSPE). similar to encephalitis but more gradual and chronic. Associated with the MMR vaccine.
SIDS and Autism - no concrete evidence showing a link.
To minimize vaccine side effects: breastfeed for at least 1-2years, minimize sugar and junk food, minimize other chemical exposure, supplement with omega-3 oil (or make sure to eat plenty of fish and eggs), take probiotics, eat fruits and vegetables, take vitamin A (proven to help immune response to infections), take vitamin C (also helps boost immunity)

Vaccine ingredients of concern:
Mercury - being phased out of many shots now but still present in all flu shots. Amounts vary by manufacturer
Aluminum - FDA states premature babies and people with compromised kidney function should not get more than 10-25mcg (micrograms) at a time. A typical round of shots given as early as the first day of life contains 250 - 1225mcg. No studies proving its safety in vaccines have been conducted though there are many other studies showing repeated exposure to high aluminum levels can cause neurologic harm.
Formaldehyde - used in very small amounts but it is a known carcinogen causing kidney damage and genetic damage
Various animal or human tissues - unknown pathogens may be present in the tissue
MSG - Present in small amounts in some vaccines. It is a known excitotoxin when consumed in large quantities or injected into the brain. Recently has been removed from many products including baby food.
2-Phenoxyethanol - Used in small amounts in vaccines. It is harmful if swallowed, inhaled, or absorbed through the skin. Can cause reproductive defects and is severely irritating to the eyes and skin. Found in many common products such as perfumes, cosmetics, inks, etc. The amounts used in vaccines have not been studied for safety.
Sources:
The Vaccine Book by Robert W. Sears, 2007 ed.
Herbal Remedies for Children's Health by Rosemary Gladstar