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No, the Spirit Airlines credit card won't allow you to receive your checked bag free. Not even $9 Fare Club membership will allow you to get a free bag. It will only get you up to 50% discount. The card will allow you to get a preferred boarding at most airports though.
Generally some airline cards have this perks, but not lowcosters. If you want to fly Spirit and save, earn rewards to purchase all your collateral expenses like checked bags. A travel credit card of any origin will allow you to do that, they are versatile. With such cards you can earn the unlimited number or miles to spend on your flights and other stuff.
If you want specifics, take a look at the Discover it® Miles. The mileage accrual here is solid, so you'll earn enough to finance your flights and all the rest. And there's no annual fee to take from your earnings. Miles never expire and there's no limit to how many you can earn. Moreover, all earned miles are matched at the end of the year (provided you are a new cardmember), meaning that you'll have twice as many miles.