A photon by definition will travel with the speed of light (assuming vacuum here; air slows things down).
To measure its speed, and thus the speed of light, have two people with synchronised watches together. They then move (slowly, or at least well below the speed of light) in opposite directions. At some predefined time, one of fires of a photon (or simply uses a flashlight). The other one times when s/he detects the light, then calculates the difference in time. Take the distance between the observers, divide by the time difference et voilĂ , the speed of light.
As long as the two people don’t move between themselves, it’s easy to measure.
(Another way of measuring just let’s the photon bounce of a mirror, back to its origin. Even simpler.)
Time slowing down is also relative: it only slows down for things moving relative to you. Everything that moves with you (train, space shuttle, next-generation space ships, anything that you feel goes fast) has the same time. So travelling in a space ship near the speed of light will not prevent you from aging in the natural way; ie, you don’t get extra years so that you can read that long list of books.