What happens if I start taking the contraceptive pill for the very first time in the middle of a period?

Use back up. First it won't work as contraception for another two to three weeks so you need to use a back up. Also on the off chance you happen to be pregnant already, you could expose the pregnancy to the hormones. Probably better to wait until your period starts.
Ok. Starting the pill during the first few days of your bleeding for a new start is best. The pill will help suppress the new egg from forming and may decrease the amount of bleeding with your cycle.