Unfortunately, there is no single, definitive answer to a question like this. All you can really do is to lay out the arguments for and against saving and make a personal judgement based on them.
There are two main arguments for starting early. The first is that saving early maximizes the length of time your savings have to grow through compounding. The second is that if you develop good habits early on, those habits will stick with you for the rest of your life. So, the theory goes, if you develop a habit of careful budgeting and setting aside a little extra while you are in your 20s, then by the time you get to your peak earning years, saving for the future will already be second nature for you.
The argument against starting early is based on the idea of consumption smoothing. At the start of your career you are earning much less than you expect to be once you become more established. The theory here is that it makes little sense for your present, relatively poor, self to transfer money to your future self, whom you expect to be relatively richer. There is a lot of sense to this argument, but of course it rests on certain assumptions about what your future will hold, and you never really know for sure if those assumptions will be borne out.
My experience was that I spent most of my 20s in graduate school, scraping by on about $17k per year. Needless to say, I didn't save a lot during that time; I thought I was doing pretty well just avoiding going into debt. I don't think the late start hurt me that much. The more important thing was that when promotions and pay increases did come, I didn't increase my spending right away, and before too long I had some decent savings going. Resisting the temptation to expand your lifestyle expenses every time you get a raise will do more to get you to financial security than ascetic living during the first few years of your career will.