If you choose to use the TurboTax online web-based editions then you will need the Self-Employed edition which supports both W-2 employee income along with personal deductions and self-employed income and expenses which are reported on a Schedule C, included with the Form 1040. If you need to also complete a state tax return that is an additional charge with the online editions and is currently $44.99 for each state program. There are no e-file fees for either a federal tax return or a state tax return using the online editions. The online editions can only be used for one tax return per account and User ID. So if another return is needed a new account would be needed to be created and paid for along with creating a new User ID for that account.
TurboTax online editions - https://turbotax.intuit.com/personal-taxes/online/
Or you could use any of the TurboTax desktop CD/Download editions that would installed on a personal computer. All the desktop editions include the same forms, schedules and worksheets so any edition could be used for your tax situation. The difference between the desktop editions is the amount of help and assistance provided within the software. So the desktop Home & Business edition would be the recommended edition in your case, but Not required.
Note also that there is one free state program download included in the purchase price of the desktop editions (except for the Basic edition). There are also 5 free federal tax return e-files included with the desktop editions. There is an e-file charge that is currently $24.99 for e-filing a state tax return but that charge can be avoided by printing and mailing the state tax return. (A New York state return is free to e-file)
TurboTax desktop CD/Download editions - https://turbotax.intuit.com/personal-taxes/cd-download/