Does more mass mean more weight?

Only if they are in the same inertial frame of reference.

On Earth if a) has a mass of 60kg and b) has a mass of 10kg - then a) would weigh more.

If a) was on the Moon and b) stayed on Earth then they would both weigh the same.

If b) decided to go to the Sun, then b) would weigh more than a).

Mass stays the same no matter where you are. Your weight is deduced by the amount of "pull" gravity has on you.