The Clerk of the Court in the county where his case was tried is your best starting point and the information is public record.
Call the clerk's office directly and ask about the case by your boyfriend's full name and date of birth. They maintain all case records including charging documents, plea agreements, trial transcripts, and sentencing orders. Most clerk's offices can tell you over the phone what sentence was imposed, and many county court systems also have online case search tools where you can look up this information by name or case number at no cost.
If you do not know which specific county handled the case, start with whatever city or county you know he was arrested in. The case would have been filed in the court that has jurisdiction over that location. A quick search for that county's court clerk contact information will get you to the right place.
TruthFinder is another option for pulling together public record information quickly if you want a starting point before making calls. It draws from court databases across the country and can surface sentencing and case information for cases in other states.
On the communication gap, if he has been sentenced and transferred to a facility, the silence is most likely a function of the intake and orientation process at wherever he landed. New arrivals typically do not have phone access for the first week or so. Once you know what facility he was sentenced to, you can send a letter through InmateAid to let him know you are trying to reach him. That letter will get there even during the orientation period when calls are not yet available.