As someone who's worked in the industry for a decade now, here's a quick rundown (US specific,) of what your schools and parents didn't teach you:
For the love of god get an account at a federally insured institution. Look for FDIC (banks) or NCUA (credit unions) insured and regulated financial institution. They are legally required to have this status publicly available and accessible so it's not hard to find.
The FDIC and/or NCUA will insure your accounts up to $250,000 PER AUTHORIZED SIGNER and per account type. These are factors to max your coverage to even higher than $250k but the key point is that if something happens to your bank or accounts there, that first $250k of your money is secured anyway.
Banks are for profit. Credit Unions are exactly what it sounds like: unions. They are not for profit and member owned.
Bigger institutions have more money and resources at their disposal; they have the fancier apps, 24/7 phone banking and more locations. But watch out! They are no different than any other large corporation you've heard of when it comes to ethics. Smaller institutions have more limitations, and lesser size is not an indicator of morality, but it's something to consider when choosing where to keep your money.
These institutions, regardless of what kind you choose, will offer interest bearing accounts. Money Market Savings and Time Accounts (also called Certificates of Deposit,) are popular choices to put the money you already have to work for you. You can earn money just for having your money in an interest bearing account type.
All financial institutions charge fees of one sort or another. They are offering products and services, after all. Nothing is free! They will also disclose options to avoid paying those fees, usually based around meeting specific criteria such as minimum balances or direct deposits.
Take this information and do your own research so that you can make an informed decision. Now you know what to look for! Don't be taken advantage of!