The first savings goal should be an emergency fund. Think of this not as an investment, but as insurance against life's woes. They happen and having this kind of money earmarked allows one to invest without needing to withdraw at an inopportune time. This should go into a "high interest" savings account or money market account. Figure three to six months of expenses.
The next goal should be retirement savings. In the US this is typically done through 401K or if your company does not offer one, either a ROTH IRA or Traditional IRA. The goal should be about 15% of your income. You should favor a 401k match over just about anything else, and then a ROTH over that.
The key to transforming from a broke college student into a person with a real job, and disposable income, is a budget. Otherwise you might just end up as a broke person with a real job (not fun). Part of your budget should include savings, spending, and giving. All three areas are the key to building wealth.
Once you have all of those taking care of the real fun begins. That is you have an emergency fund, you are putting 15% to retirement, you are spending some on yourself, and giving to a charity of your choice. Then you can dream some with any money left over (after expenses of course).
Do you want to retire early? Invest more for retirement. Looking to buy a home or own a bunch of rental property? Start educating yourself and invest for that. Are you passionate about a certain charity? Give more and save some money to take time off in order to volunteer for that charity.
All that and more can be yours. Budgeting is a key concept, and the younger you start the easier it gets.
While the financiers will disagree with me, you cannot really invest if you are borrowing money. Keep debt to zero or just on a primary residence. I can tell you from personal experience that I did not started building wealth until I made a firm commitment to being out of debt.
Buy cars for cash and never pay credit card interest. Pay off student loans as soon as possible.
For some reason the idea of giving to charity invokes rancor. A cursory study of millionaires will indicate some surprising facts: most of them are self made, most of them behave differently than pop culture, and among other things most of them are generous givers. Building wealth is about behavior. Giving to charity is part of that behavior. Its my own theory that giving does almost no good for the recipient, but a great amount of good for the giver.
This may seem difficult to believe, but I ask that you try it.