I have iMac Most recent macOS Sierra 10.12.6. I need to update to at least OS 10.13.6. Is this possible? TIA

Thank you. You can run OS 10.13 High Sierra but no higher. We have a 2011 that runs 10.13 without issue. Spending money on it has to include consideration of its not supporting newer app and macOS versions.

Is this a 21.5-inch iMac? If so, any slowness you have with slow startup and slow app launching is NOT a lack of RAM but rather an under-spec mechanical hard drive Apple opted to install in the 21.5-inch versions to keep the price down. You can throw massive amts of RAM at the hard drive problem and see no improvement.

Because I had compatible RAM removed from another Mac, I upgraded our 2011 21.5 iMac to 8GB RAM. It did not make any difference--startup speed and app launch were unchanged.

Were I in your situation I would see how it ran with 4GB RAM first, noting what is slow. Remember, starting with macOS 10.9 Mavericks, Apple redesigned their RAM management to stage unused RAM for faster access. That favors a device with low RAM.

There is a simple plan to run High Sierra on 10.13 with 4GB RAM, at least for a trial period:

• Don't install any anti-virus or so-called "cleaning" apps.
• Don't used Chrome. It likes to hog resources.
• Simple restart the computer every 2-3 days to flush the RAM.

EDIT: Increasing RAM from 4 to8GB is about US$20; going to 12GB is about US$45. I get our RAM here:

https://eshop.macsales.com/shop/memory/iMac/2011/DDR3_21.5_27

As an owner of the same model and having a lot of hardware upgrade experience I can say that the only mindblowing speed difference would be to install a new internal solid-state drive (SSD) rated SATA 6GB/sec. The startup and app launch time would drop to 1/4 to 1/5 of current based on my upgrades of a similarly-spec Mac. However, that would cost around US$100 for a 500GB SSD if you do the work yourself. Add another $75-100 if you need professional installation.

Any Mac capable of running macOS 10.12 Sierra can run macOS 10.13 High Sierra.

Installers for older version of macOS are in this Apple support article:

How to get old versions of macOS - Apple Support

Two this to note: 1) use only Safari for macOS system downloads, and 2) if the installer says it is "damaged," that only means the installer found and older version of itself residing in your Applications folder. Remove to old one, then proceed.

Only your iMac sub-model will tell us if you can run higher than 10.13, and we do not know your sub-model. Please do "About this Mac" from your Apple menu to get this screen:

Please post the sub-model but NOT the serial number. With that we can see if you have a different upgrade path available to you.