What kind of breaker do I have to use for a sub panel of 125 amps? May I use a 125 amp breaker (same amp rating as the sub panel) from main panel to feed the sub panel?
The breaker size is decided by the wire connecting the panels. For instance if the wire is 8 AWG, the breaker will be 40A. Regardless of the wire size, the breaker cannot exceed 125A, presuming that is the rating of the panel.
As always, make sure you are buying a subpanel that is large enough. It is very common for someone to save $40 on the panel, only to later have to spend a lot more than $40 for twin breakers and other work... because they got a panel that left them too little room for expansion. Panel space is cheap.
It is totally fine to have a subpanel whose "main breaker" is larger than the breaker in the main panel serving it. For instance I have a 60A circuit serving a too-small 4-space subpanel. I replace that with a 42-space subpanel with a 225A main breaker. That is fine. The big main simply serves as a shutoff switch. (And I made out because the bundle came with 22 ”bonus breakers" which I use elsewhere.)