Why does your lawnmower make a grinding noise when you start it?
If the mower has an electric starter, the gear on the starter is worn. Replacement is not hard and the part (a plastic gear) is relatively inexpensive.
Why would my car make a grinding noise when i start it and after running for a bit the grinding noise becomes slower?
Don't ignore that kind of sound. Find out what is causing it and replace or repair the faulty part. A "grinding noise" can be caused by an alternator, water pump, idler pulley, heater fan motor, air pump or any of a long list of components, many of which can cause some serious problems if they seize or come apart.