Can you fly business class for free you ask? Well, of course you can! There is a caveat though...Your credit score you be in the 700+ range. If you don't yet have credit or are in the process of building credit I would recommend getting one of the basic credit cards out there with no fee like the Bank of America cash rewards card:
Which has no annual fee and gives you 3% on a category of your choice and 2% on other categories. Another good one would be the citi double cash card.
Or if you like staying at hotels for free you can opt for the Hilton Card which has no annual fee and gives you 100k bonus points after spending $1000 in the first 3 months.
Use these cards and pay your balance off in full each month. Keep credit utilization to a minimum and you should see your score go up. But once you graduate from credit school and have a 700+ credit score, you can start getting the cards with the big sign up bonuses. These cards usually carry an annual fee, but they offer a high sign up bonus upfront to make up for it.
Some examples are
The Chase sapphire preffered:
https://www.referyourchasecard.com/6a/SBQHOIWRCL
Right now gives you a 60k sign up bonus after spending $4000 in the first 3 months and comes with a 95$ annual fee. The fee is waived the first year, and after the first year comes you can downgrade it to another card to avoid paying the fee.
The Amex platinum
It gives you 60k bonus after spending $5000 in the first 3 months and comes with a steep $550 annual fee. They make up for the fee by giving you $200 in credits for airline fees, $200 in uber rides and $100 i saks gith avenue.
How do these cards get you free business class tickets? I'm glad you asked...Those sign up bonuses you get after opening the card can be transferred to airlines who partner the the cards and converted to miles. For example, AMEX partners with Delta which means you can transfer your AMEX points to your delta skymiles account(You can open one for free if you don't already have one) and use those miles for travel.
An example would be transferring your AMEX points British Airways avios miles and flying from the east coast to Europe for 34,000 miles. You can even find travel partners that let you fly from the US to Japan on business class for as low as 80,000 miles!