First bike tip: PAY ATTENTION WHEN YOU'RE ON A STREET. Pay attention to everyone, including walkers, bikers and roller bladers (outside of town). You would not believe how many people do not pay attention to the mass amounts of people on the streets, especially walkers, who will cross the street without even stopping for horse-drawn carriages. Accidents in general are shockingly common along the main streets of town, and every year people get seriously injured.
Second bike tip: You'd think your bike is safe because there are literally thousands on the island at any given point, but please, lock your bike (preferably off the road) if you're out before ten am or out after about 8 pm. People will sometimes steal bikes for joyrides and then dump them on the side of the road. The police will even confiscate your bike if it's left on the street overnight, because of the streetcleaners. I lost my Gary Fisher for three months before the police department found it (I had a license, since I was living on the island). The lock doesn't have to be expensive, and you don't even have to lock it--just get a chain and loop it around the handlebars and tire(s). This is usually enough to avoid theft.
Whenever the roads are wet, whether from a recent rain or last night's street cleaning, avoid riding a bike until the roads have dried. Since horses roam the island freely, road apples are consequently everywhere-- and when it rains, the roads become uniformly (for lack of a better word) dirty. Riding a bike, even if it has a tire guard, will send all of this road grime up onto your back and rear, and by the end of your enjoyable bike ride the entire back of your shirt will be brownish black. If you're going downhill, you may be unlucky enough to get it on your front-- and as someone who has had this happen--it's not pleasant! This tip is a little bit gross, but if you want to avoid being covered in 'recycled hay' and being the butt of local jokes, I suggest either wearing a rain coat or waiting a few hours for the roads to dry.
Swimming on the island is great fun, but watch out for signs posted about currents. At a few points on the island (like Windemere Point) there are conflicting currents that can carry unattended children or objects away. I can attest to this fact: I once watched my soccer ball get carried out into Lake Huron just by the currents a few summers ago. I never saw it again.