Reactions can vary from a swelling at the vaccine site to vomiting and diarrhea, welts and swellings of the face and skin to full anaphylaxis. These can often be managed with benadryl and other drugs if they occur. Since most of the diseases are important to vaccinate for we must limit reactions without limiting vaccination. The first step is to eliminate unnecessary vaccines. Call your vet if one does.
Different vaccines use different adjuvants.
The adjuvant is frequently the cause of the reaction. Also you may want to leave the pet at the vet for the day of the vaccination so they can observe the pet and help if necessary. Give the vaccines on different days at different sites to limit the reactions. I should also comment on long term risks of vaccination. Recently in studies of fibrosarcomas in cats vaccination was suggested to predispose cats to these tumors.