Car & Motorcycle
The best way to enjoy the trail, with the freedom of starting and stopping as you choose, is by self-driving. Most folks rent a car in Sydney or Halifax and drive across, although it's worth noting that car-rental agencies charge high one-way fees for returning the vehicle in a different town, so it's a good idea to plot your route in a loop.
The next decision is which way round you're going to follow the trail. The vast majority of drivers tend to do it from west to east, starting in Chéticamp and ending in Ingonish, but there's a good case for doing it in the opposite direction: there's less traffic and you get to drive coast-side the whole way, making for better views and easier stops. It also means that, if you do the trail in a day, you'll hit the west coast for sunset: cue photo ops aplenty.
Note that there are very few petrol stations along the way: make sure you fill up in Chéticamp or Ingonish before you set out.
You'd need thighs of steel and the heart of a lion to cycle the Cabot Trail, though of course it has been done. Heavy traffic in peak season is another reason that this is not a recommended plan of attack: it's for hard-core cyclists only.