Now that you have officially mastered the art of saving cash in a jar on your desk, it's time to learn about saving cash in a more secure location. Some other reasons you might want to use a bank: your jar of cash is overflowing with money, you want more security, you don't typically carry cash so it's hard to save it, you want a more automated savings process, etc.
If you don't yet have a bank account, you should really open one. I recommend using a major financial firm for your primary checking account. Some examples include: Chase, Bank of America, Wells Fargo. These are full service banks that have branches all over the country and can meet any of your banking needs (physical check books, facilitate a mortgage, a large network of ATMs, etc). If your job pays with direct deposit, I would suggest having that money deposited to your primary bank.
I would also suggest creating a savings account at a secondary bank where you can find a much higher interest rate. If you remember, an interest rate is the amount of money the bank pays you for giving them your money. While Chase gives you 0.01% per year, Goldman Sachs Bank gives you 1.70% per year. Yes, that is 170x more money and that's a lot of dog food!
You get the higher rates at Goldman Sachs because they don't offer any of the traditional banking services your primary bank will do (no physical locations, no ATMs, no paper statements, etc).
Once you get your secondary account created, you can link it to your primary account, and automatically move money from your primary account to your secondary account. This forced savings coupled with the much higher growth rate will help you save money more quickly.
Remember, deposits in all banks are insured by the federal government up to $250,000 per account holder. An individual account is insured up to $250,000 and a joint account is insured up to $500,000. If you have more cash that you want to deposit, just open a new account at another secondary institution. Some examples include: