Before you save for anything else, you should first build up an emergency fund that you can fall back on in case something goes wrong, this should be in an easily-accessible savings account.
Things you plan to do in short term
For money you'll need in the short term, perhaps for a deposit on a house, or buying your first car, saving makes sense because your investment may not have enough time to make up any fall in value.
Things you plan to do in medium term
For medium-term money, maybe to pay for a child's wedding, saving could make sense – although investing could earn you a greater return on your money.
Things you want to do in the long term
For money you're not going to need for years, such as for a place to retire to, investing could earn you a greater return – because with saving the value of your money will get eroded by inflation over time.