I'm not sure what exactly you mean with "saved card lists".
There is no way to make a list of cards and then make bot games that have kingdoms that consist of random cards from this list.
There is also no way to "store" kingdoms you've played or designed and then "load" them from within dominion.games.
The latter option can be realised somewhat because the game allows text input for kingdom selection. In the picture below I created a kingdom with some cards, by selecting them one by one in the "My Table" tab (red). The kingdom is represented in text form below the picture (blue):
You can select and then copy that text with your mouse like you can any other text in a webpage. If you store it yourself on your device, you can then later paste it into the following text box, allowing you to replay that kingdom:
So you can save and store kingdoms, but you must "provide" the storage space for yourself.
A built in way to allow players to gradually practice with more and more complicated abilities is the card pool level. Card Pool level 1 is the base set only, level 10 is all existing cards, and in between it introduces cards based on the kind of effects they have. There are cards in each expansion that are mechanically simple, so they might be easier to learn than some of the cards from earlier expansions (like Duration cards).
You can adjust the card pool level in games on your table using this: