Hi J. B,
Yes, this is completely normal and there is no single "correct" way to handle it. You might gather ideas to try from all these great posts and from your own instincts. Choose your favorite and use it CONSISTENTLY.
For example, you might try "floor time" for the baby and "Mommy / (older son's name here)" time for you. You and your son can brainstorm some sit-down activities for you and the three year old to do while the baby is on the floor. Get your son to notice what the baby is doing while you sit together. At other times in the day, the 3 year old could do some activity by himself (which encourages independent play), while you monitor him and the baby.
Another suggestion that worked great when my kids were young was to lay down chairs, cushions, etc. to block off a safe area for the baby to play in. Be sure to child-proof it and put some of baby's favorite toys in there. You can invite your 3 year old to help locate it and build it. Then the 3 year old should ask before going in with the baby and then only with your close supervision.
Do you get the 3 year old to help (check on the baby, get diapers, comfort baby, etc?) All of this helps instill the love of a big brother and can turn the mind to that of caregiver instead of pain-giver.
Your instincts are right in keeping both kids safe from harm. You cannot allow your 3 year old to think rough play around the baby is okay. So if the fun or "good time" ideas don't work, give a firm and reasonable consequence. To some degree, it sounds like your 3 year old is testing your resolve to keep him from being rough with the baby.
In all areas, if you are consitant in whatever you do and in the limits you set, your 3 year old should settle down and listen.
I'd love to hear how this goes.
Take care,
Parent Coach J. B