Although watching broadcasts and even YouTube guides is a solid way to start, there is no going around the fact that the fastest way to learn something new is to do it yourself. Luckily, three of the most popular esports titles in Counter-Strike: Global Offensive, League of Legends and Dota2 are all available for free, meaning anyone can get familiar with the basics of any game without spending any money.
Additionally, Dota2 and LoL are both MOBA (Multiplayer online battle arena) games, so understanding one will help you get the grasp of the other significantly faster.