Games won't install to the system like PS4 and XBO. So if you play a cart-based game, the game stays on-cart but the save file exists in system memory.
So if you rent a game, play for a bit, return the game, then choose to buy it, you don't have to start over. You can continue from where you left off.
There is also an eShop that lets you buy games digitally (and I would recommend having an SD card if you buy a lot of larger games). There are a ton of smaller, cheaper, indie-created games on the system. There are already almost 350 games available to buy on the game, which is insane for a system that has been out for only 11 months. So while they obviously won't be as good or as long as the Zelda or Mario games that are $60 each, you should be able to find a few $8-20 games in there that may be of interest to get some content on the system until you're able to buy some of the larger games.
There are also 21 games with demos that you can download for free, which will give you something to play and see what you like or don't like when choosing games to buy.
There are two pinball games on the system that is free to download and comes with one table, with the ability to purchase additional tables within the games. So that is more content that you can add for free to see if it's a game you like.
Hope this all helps. Enjoy the new system!