Glad to hear you are starting on building your financial base early. In addition to what's been said regarding savings, is to have goals for your financial base. There are different reasons to save money - emergency backup, vacation, house/car, retirement, health, etc.
I recommend your first savings goal is to save up a 6 month emergency fund. The emergency fund should have enough money in it to cover 6 months living expenses - rent/mortgage, utility bills, vehicle maintenance (including gas and insurance), food, other incidentals, etc. Take time to calculate how much you are spending each month - including fun activities, and determine what you need to sustain yourself for 6 months. That will be your beginning target savings amount. The purpose of this is to provide some financial security should something happen where you cannot earn money - lost job, injury causing you to quit, etc - to help sustain you while you come up with plan B.
Once you obtain this goal, then you start thinking of the next goal to save for in addition to those funds - car, house, retirement, HSA (health savings account), etc.
Another thing if you haven't done this yet - now that you have both a checking and savings account - if they are at the same bank - if you haven't done so, you'll want to include overdraft protection on your checking account. What this does is if your checking balance does ever go below zero, the system will withdraw money for a smaller fee than if your checking account bounced the check for insufficient funds. This typically will happen at some point in your life when finances get tight, so having this gives you some financial protection by reducing fees in such circumstances.
Lastly, when things start to stable out for you, you might want to consider looking for banks that offer Reward Checking accounts. I started researching those as alternative short term "savings" accounts for myself last year. These accounts will pay a significantly higher interest on your balance than a traditional savings account (right now with these low interest rates, my traditional savings account is only paying 0.1% interest, but my reward savings is paying 2.01% interest - big difference). To really leverage a Reward Checking account takes some discipline. You need to think of it more as a savings account vs checking account. Typically these accounts require a direct deposit, a certain number of signature debit transactions (where you pay with your debit card, but not using your PIN), and getting your statements electronically. Where some people fail to leverage such accounts is with the debit transaction requirement by overspending. When I setup mine, I budgeted how much I would spend vs deposit, and am using this account to pay most of my monthly, low dollar bills, which leaves a handful of transaction I can use for things like eating out on the weekends. I recommend before setting up a Reward Checking account, that you plan out your strategy of how you plan to minimize your spending so your balance in that account grows. I maintain that advantage by keeping my traditional checking account (at a different bank) for the day to day spending so my Reward Checking account can remain fairly untouched.
Hope these give you some ideas for financial goal options.