You need to spend 10$ in order to earn rewards, everything else is basically free. You can invest more money if you want to and this will definitely speed things up for you but I wouldn't advise doing so before you understand the game a little better, otherwise you risk buying cards you won't really need later on.
Regarding the actual game - Splinterlands is an auto battler so you only chose the cards you want to play and their order. There's no back and fourth like in Hearthstone. Because of this the game is a lot less complex but each round is also played considerably faster.