Zelda BOTW is the best game for the system, at least in my opinion. It might be a little complex for a six year old but you'd enjoy it. Mario Odyssey is good too. It's very easy and is probably a little more accessible for a younger kid than Zelda is.
All games can be downloaded nowadays. You never have to actually go to the store to buy them. With that being said, if you download games, or have a lot of games, you'll want to upgrade the system storage with a micro SD card. There are a lot on sale for Black Friday so it's not a bad time to get one. You don't need the Nintendo branded one.
I buy the big titles like Zelda, Mario, etc on cartridges. For small games or non-nintendo games I download them. If your kid is going to play on the go a lot, downloading has the advantage that you don't need to bring physical games with you.