When researching service histories of those who enlisted into the Royal Marines, it should be noted that, unlike today, in WW2 not all Royal Marines were Commando trained. The first Royal Marine Commando unit was not formed until February 1942, over 18 months after the formation of the Army Commandos. 

Royal Marines service records are often quite scant in specifics, however unless you apply for it you will not know what it contains. It really is important to apply at an early stage in your research. How to do so can be found on the link below titled Service Records and Medals.

Royal Marine Commando units of WW2 were numbered 40 to 48 (with one or two exceptions), and were part of Special Service Brigades (later renamed Commando Brigades), along with their Army Commando counterparts. However there were many more non commando Royal Marine units in WW2.

The following link takes you to a full list of units of the Royal Marines from 1919 onwards

[Royal Marines Unit History Summaries].

This link will take you to a list of official documents relating to the Royal Navy and Royal Marines

[RM Historical Files].