Things have changed a lot in recent years that sort of make the savings account no big deal. The rate of interest is extremely low. Also, some accounts may have a limit on how many withdrawals you can make from savings in a month, without incurring a fee for the extra withdrawals.
However, just starting out it could be a good idea for you to have one, at the same bank where your checking account will be. What you should do is determine how much you want to put aside into savings each month. Treat this just like you would a car payment or rent; make the deposit every month or every paycheck. Don't cheat; always pay yourself like this as you pay the other bills. Eventually, you will have a tidy sum there and you can then look at other investments where the return is greater.
Choose a bank that has the lowest (or no) fees for the account you are opening. Often this will be found at a Credit Union and not a commercial bank. About the only downside to a Credit Union, for someone at your stage, is that they may not have as many ATM locations that are convenient to you. But if you use checks and a debit card and don't walk around with a lot of cash, that may not be a problem. In emergencies you can get cash from any ATM and just pay the extra fees for the withdrawal that one time.
Most financial institutions require that you open a savings account before you open a checking account. I would recommend a credit union for opening an account, because they usually have lower (if any fees) on their accounts, unlike a bank. Be sure to make sure that you have overdraft protection on your checking account or that it's set up that it can't go negative. I've seen too many people get into financial trouble because of overdraft fees.
you need to have a savings account. employers need an account to deposit your earnings into on payday as issuing payroll checks separately is becoming scarce. select a bank close to home. most savings have a minimum amount to be maintained at all times to avoid a monthly fee. frequently, with an existing savings account they will also offer a free checking account. watch, some checking have mandatory monthly fees. this can be costly.
I bank at a smaller bank Regions, I like it. the choice is yours. You can open a checking account and later a saving if you manage to be able to add money to saving. Always good to have saving in case of emergency.