If you have two different types of tires on the front and back of your car will it effect the handling of your car?

It probably will. Each brand and type of tire has it's own ride & handling characteristice. Depending on the tires the difference may take a professional evaluator to feel it or "even a caveman" would fee it if you were to put High Performance on one axle and snow tires on the other.

You may not have an issue if you keep same speed rating on all 4 tires and same general type such as Snow or All Season or Summer.

Read the sidewall and see if the construction materials are the same too.

NEVER mix radial and bias (diagonal) on a vehicle. Use one construction or the other.