League of Legends rotates through a group of champions every week, which you can play without unlocking. Every week will be different, every week will have you trying something new. This is a great way to try out a champion before you actually commit to buying it, and a good way to figure out what you like to play. This also lets you build up enough Influence Points, or IP, to buy the champion you really want.
Using the "free to play" rotations for the first couple of weeks, I learned which champions I liked to play, and which ones weren't so fun. I played the tutorial like everyone else, learned how Ashe works, and then started in on the realm of PvP. After fiddling around for a while, I settled on Mordekaiser, running him as an AP mid, based off of the guides from MobaFire. Once you pick a champion you like, it's very easy to not spend any IP for a significant amount of time. Players who like to have a variety of champions, well...then things get a little tricky for them. There's a long grind ahead to get enough IP to buy all the champions that are currently available. Of course, there's always a faster alternative...
Riot Games will try to get you to buy riot points every chance they get. They post all sorts of enticing offers and throw everything they can at you to get you to be a paying customer. Fortunately, You can get away with not spending any cash to play this game. While it might expedite your progress and give you some awesome items in return, it's not a necessary thing.
Yes, you can play League of Legends for free. It's difficult, but it can be done. Give it a go, if you haven't already.