My father and I are thinking about opening our own cleaning business. My question is do we need to purchase insurance and bond? Or either one?

Does any of the clients you are going to be working for require all of your workers to be bonded, Are you going to require your own workers to be bonded for handling your money?
Normally you want to have liability insurance, you will need commercial insurance on your vechiles you use, if you have equipment you may want to get a rider on them on your home owners for parttime business, or a separate policy. If you have workers you will need workers comp insurance, you should also have accident insurance on yourself ( since owners are not covered by workers comp)