first grade is a very tough adjustment for many children, particularly boys. The symptoms you are describing could reflect ADHD. They sound like the typical descriptions I hear from families of children who meet diagnostic criteria for the disorder. There is research to suggest that ADHD is heritable, however that does not necessarily mean you child has it because some relatives do, just that he is at higher risk.
You are correct that this is still early in the school year, though keep in mind that most children have a 'honeymoon' for the first month and then their behavior declines. I would ask the teacher how she is attempting to manage this behavior, and keep in close communication with her. I would not wait until things get worse before asking for help. Ask the teacher to set up a team meeting with you, other teachers, and school staff such as an administrator to make an informal plan for how to help this child. If the plan does not work (and you will know in a week or two if things are getting any better), then I would recommend you seek a psychological consultation.
Psychologists can do 2 useful things for you--they perform psychological assessments and can teach you how to manage your son's behavior. You will not know if your son has ADHD until you have a proper assessment done (see my Medhelp article on ADHD), though even if he does not, you will still need help to get this behavior under control. An assessment will yield data that will help inform what interventions will be helpful at school, such as a behavior plan, accommodations or even special education services if necessary. The book Dr. Larry Silver's Advice to Parents on ADHD is a great resource for learning more.