Always let off the throttle when you shift. You can take off in second as long as you're not going up a steep hill, pulling something, or have a lot of weight on the ATV. First on those ATVs is actually super low. If you try to take off in higher gears than second, you can damage the centrifugal clutch.
On the Fourtrax 300, you don't need to take the hub off to change the brakes, but on yours, the Fourtrax 350, I think you have to take the hub off to change them.