seeking help from apple support. when i put my correct serial number in, i am told that it is not correct.

Apple system serial numbers have 12 characters, and accessories like keyboards and mice have a longer 17 character serial number. If your serial number is longer or shorter than 12 characters then you're not going to have much success getting support via the online system until you can locate the correct 12 character serial number. Also, remember that Apple uses zeroes, not the letter 'O' in the serial numbers, so if you're entering the letter 'O' instead of a zero that could cause issues with serial number recognition. In many cases I simply select 'About this Mac...' under the Apple in the top left corner and then cut and paste the serial number from the dialog box that appears.

This article provides links to product specific serial number locations.

Find the serial number of your Apple product - Apple Support

You don't require any serial number. You require an Apple ID (an email address) and a password that you use with your Apple ID.

The only other thing the computer will ask for is your Admin account username and the password you setup for it. This is the login account on the computer.

If you need the actual serial number of your computer, then click on the Apple menu and select About This Mac. The serial number is usually at the bottom. Copy it because you need to use it exactly as it appears.

To upgrade to Sierra:

Your iMac must be a Late-2009 or later model.

You must have Lion or later thru El Capitan already installed.

A minimum of 2GBs of RAM (in reality you should have a minimum of 4GBs of RAM.

At least 20GBs of free space on the disk drive.

If you have Snow Leopard 10.6.x installed, then you must be sure it's upgraded to 10.6.8 or Lion 10.7.5 in order to run Sierra.

If you cannot meet the above then El Capitan may be as far as you can go or Lion may be the limit.

Lion System Requirements

1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7,

or Xeon processor

2. 2GB of memory

3. OS X v10.6.6 or later (v10.6.8 recommended)

4. 7GB of available space

5. Some features require an Apple ID; terms apply.

Are my applications compatible? See App Compatibility Table - RoaringApps.

If you have an older Mac and really have OS 10.6, your serial number may not be in a current database. If you try to call Apple, explain that your Mac is a (year/model) and is running 10.6. If you call, you can also try to enter random numbers and letters - the system will tell you that it does not recognize it and will hopefully connect you to someone anyway (they did when I accidentally entered the wrong serial number).

And, look at Kappy's post for requirements to update. If you are running 10.6, you need to update to 10.6.8 first. Once you do that, you will have the app store (in Applications) which you need to use to update further - that is if your Mac will support it.