The tax amount will go down because the biggest component is based on the assessed value which will be lower today than if you bought the property yesterday. If you already own property and want the County to lower your assessment, you can get the forms from the County and they will review it.
I'm helping several clients with that and would be happy to do the same for you or others.
Many of the other charges have different bases and formulas for assessment. There are two ways to get that information. (1) You have 17 days after escrow opens on a home to review relevant disclosures, one of which includes a detailed review of taxes; or (2) with a little research you can review tax bills for particular parcels online.
Again, I help many clients with that and would love the opportunity to compete for your business too.