Many people have suggested that I start a savings account as soon as possible since I'll find it really helpful later on in life, but most of them just give me vague answers or tell me to go visit a bank and ask for information when I ask them how to do that, so what can I do to start one?
You've been given the best advice. You know why they tell you to go to a bank? Because they specialise in money at banks. You just need to go and open up a savings account and start making monthly deposits. Be strict with the money you are putting in and never cheat yourself.
If you are simply looking for a savings account, then a bank is fine. Or you can open an online banking account with a site such as ally. If you have decent extra money to invest find an investment firm that takes only small fees and get into a money market type account.
Most banks offer banking and investment services. Since what you're interested in is investment and not just saving [since the interest earned is won't make you rich anyway] best thing is to ask the bank how they'd best invest your money for you. They'll offer suggestions. . .