I go brown really easily without burning - does that mean I can forget about skin damage or cancer?

Unless it's out of a bottle, there's no such thing as a "safe" tan. The brown colour is a result of melanin, which is produced in larger quantities when the skin is exposed to UV radiation. The more melanin that is produced, the darker the skin will go. So this over-enthusiastic production of melanin is just the skin's way of protecting itself against UV damage. Just because you don't burn doesn't mean you're not damaging your skin. If your skin turns brown easily, it means it's producing more melanin, and while that may offer a bit of protection, sun damage is still occurring.