what kind of breaker for subpanel?
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.)