Having no idea what your total income was (it doesn’t matter how much you make per hour, after all, if you work less than full time), it’s hard to say “you’ll get everything back”, but if you fill out the form honestly (and correctly, but it’s a pretty simple form! I doubt that you’ll have much problem with it), then you’ll get everything back that you should get back.
My strong recommendation to anyone who is starting their working / taxpaying career (and congratulations to you for doing that!) is “compute your W-4 as carefully as you can to have the least tax withheld – within the limits of the law – throughout the year.
You’ll end up having to pay tax at the end of the year (hopefully!), but it shouldn’t be much: under $100 or so. And you’ll have the benefit of knowing that no one else is holding money that should be yours; you won’t have to wait for it; you won’t have given an interest-free loan to the US government for 9–12 months or more. As you get more experienced at this, you may get to my ideal state: filing a tax return a week or two before it’s due and owing about $25 – $50, total. That makes paying the extra tax pretty easy to do, as well.
And then learn to live – and live well – within your take-home income, and save some so that paying your nominal tax bill at the end of the year isn’t a huge headache, and you’ll start to build an additional nest egg next to the 401(k). (Congratulations on having the sense to do that, too.)
I expect that NY tax forms are also available at the Post Office. Libraries and town halls are good places to look, as well.
TuboTax is definitely “easy”. I suspect that in your case it is also unnecessary. The program itself (especially if you buy it with the state tax adder) costs around $45 or so. TurboTax may offer a “free online” tax prep to introduce new users to the system, so definitely give that a look. But I think that their software, good as it is, offers a lot more to its purchasers than you really need. Your return should be quite simple.