Keep in mind that you very nearly gain more HP from Assist cards than you ever will from level ups, so it's important to keep up with them. The better the card, the better the HP.
The primary means of getting Assist cards is through chests, and 1/4 or so of the time a Moogle will pop out. This Moogle can give you a card for free (the bottom-right button, supposedly this is you giving him an "Unripe Kupo Fruit" but I dont't know where you keep pulling the things because you never run out). This card has better odds of being a high-rarity card than the chest normally offers. But the big point of the Moogle is to give them Ripe Kupo fruits. Those are a lot harder to find - right now you can only get them through the ongoing damage contest, who knows what it will be next. Ripe Kupo Fruits have much better odds of getting high rarity card, like SR and SR+ cards, though they might still give you Rs and disappoint you. You can exchange 1 for 1 card (the top button) or 3 for 5 cards (the middle), but getting three will take a while.
Lastly you can currently buy some Assist Cards in the Mog Shop but availability changes.