Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
It is true, you need Mac OS X 10.7 or later to upgrade to Sierra. Good news is, you can download 10.11 El Capitan first and then upgrade to Sierra.
You can download 10.11 El Capitan from here --> https://support.apple.com/en-us/HT206886
or click this link --> https://itunes.apple.com/app/os-x-el-capitan/id1147835434?mt=12
This will take you to App Store with 10.11 El Capitan link. So click GET button and use your Apple ID and password ***** download it. Once it is downloaded, it will open the installer (will take a few hours to download) and follow the onscreen instructions to install it.
Note : Although upgrading won't delete any files, I'd recommend to backup your important files to an external hard drive or any other disk. Also, you will no longer have iPhoto in latest OS X. Instead you will see a new app named Photos. So open it and it will load the photos that were in iPhoto, if any.
If you have any questions or would like any additional information, please don't hesitate to ask.
Note : This is not my final answer. I strive to provide Excellent service or 5 star service. If you feel you've received anything less or if you are not satisfied or if my solutions didn't work, please don't rate me bad or poor or ok. Instead, please let me know and I will try my best to help you until you are fully satisfied or the issue is resolved.