Doctors will often recommend waiting a month for condomless sex because the most common time for people to quit taking the pill (oftentimes due to side effects like breakthrough bleeding and cramps, not being able to stay in the routine of taking it, etc) is in the first month. The rationale is that after a month of taking it and being in the routine you’re less likely to compromise your protection by missing pills (because you’re in the routine of taking it daily) and less likely to outright stop taking the pill.
Check your pills pamphlet for details on when it protects you. Most combined pills are 7 full days (meaning protected beginning day 8), but there are a few that take longer (if I remember correctly, there’s one that takes 9 or 10 days, and then one that is packaged with placebos in week one so I takes 14 days).