Boot Camp, have you attend one yet?
To directly answer your questions: yes, we have basic training, but it is extremely limited when compared with federal service. Our basic soldier school is 3 days long. We basically learn important customs (such as when/how to salute), how to do basic drill, and chain of command. We can't be sent to federal training, by the MASG, as the Constitution of the Commonwealth of Massachusetts, which provides for our existence, doesn't allow us to be deployed out of the state.
We do not have MOS schools. The MASG is very small, right now, and isn't really big enough for MOSes, so we're pretty much all generalists (though all positions get OJT). People receive TDA positions based on their existing skill sets - we're all volunteers, we all work for the flag, and we all have civilian jobs and education that are heavily relied upon. However, if there is something specific that an individual wants to do (like be a clerk), the officers make every effort to give them that opportunity - based on seniority and qualifications.
All NCOs and NCO candidates have attended training from the RTI at Camp Edwards, over the course of 2 days (it was customized for the MASG). Our NCO corp has a bunch of former federal service NCOs, including our CSM, SGM and 1SG.
I don't know the details of officer training, though our officer corp is 90% or more former federal military officers.