A learner's permit for regular car driving in Oregon means you have to be 15 years old and pass a written test. You must have a licensed driver 21 or older in the passenger seat at all times, and can't have any passengers that aren't immediate family members. Once you turn 16 you can get a provisional license, which means no passengers for 6 months except immediate family members, no driving between late night and really late night (I think it's 12-5 am) for the first year. The provisional part expires 12 months after licensing. If you're 18 you can skip the learner's and provisional providing you pass the written and driving tests for the license.
Don't know about permits and bikes (so I guess my reply is kind of worthless

), but I know that with a license, provisional or normal, you can ride any size of street legal motorcycle if you pass the written and driving tests for a motorcycle endorsement. By taking a safety class (MSF or Team Oregon), there are no DMV tests. You just bring your certificate and money to them and they'll print you a motorcycle endorsed driver's license.
So as a simpler response to your question, most if not all states have no restrictions on engine size.