Property taxes: Check with your local city or county tax assessor's office. They will be using a formula--often it's assessed value times a tax rate. For example: Assessed value x 1 cent per hundred dollars assessed value. So if the house is assessed at $500,000, then it's $500,000 x 1 cent (per hundred dollars assessed value)=$5,000 in taxes. It'll be easy to find out what the property taxes were last year. They're a matter of public record. Obviously, both assessments and the tax rate can be adjusted yearly, but generally not by too much. Your Realtor should be able to tell you whether the assessment is likely to go up or down (generally based on what you're paying for the house), and you can find out whether the tax rate is being adjusted.
Homeowner's insurance: Just call an insurance company or two. Your Realtor probably will have some suggestions. But if you already have homeowner's insurance (or renter's insurance), start with that company. Insurance rates will vary some, so call 2 or 3 companies.
Hope that helps.