This is a top level overview of steps to provide a general understanding. This helps with understanding how to solve problems when the upgrade process encounters roadblocks.
This upgrade effort require opening up the iMac and do careful surgical removal and install. Flashing firmware to the GPU card AND install patched macOS (make migration much more difficult)
It is possible to avoid making any hardware changes to the GPU and just run the patched macOS to use key features like more modern browser. Non metal GPUs best to stay <= Big Sur with most functionality. This require installing OCLP which still require some computer savvy. But no hardware modifications are required.
https://sites.google.com/view/imac-metal-gpu/oclp
This is an over simplified diagram of all the pieces involved. Maybe worth reviewing as you learn more pieces on just how this whole modified system is put together. Hopefully it also provide a sense of why we have to patch X and flash Y.
https://drive.google.com/file/d/1B9AxmhIFQe2QGh6dmUa8dj7cW-h9YFRP/view?usp=sharing
Necessary to clearly identify your iMac Model ID. Model ID is a basic internal hardware architecture. Different model IDs are different internal architectures EVEN if externally looks identical. Model ID is critical to understand how to make decision on metal GPU card availability and flashing the PC metal GPU card to make it look like a Mac GPU card.
everymac.com provides a lookup feature based on Mac's Serial Number
https://everymac.com/ultimate-mac-lookup/
Everymac also provides a complete listing that is mostly chronological https://everymac.com/systems/apple/imac/index-imac.html
This is a general overview of steps. All steps here have a detailed section in this website to go into more details.
Note there are few guides online, most have some out of date info (software tools are evolving quickly), not quite the simplest method, and generally don't tell what to do in error conditions (not rare) Generally the newer the guide is, the more accurate the info.
There are also other possible order of steps. This ordering assumes very little is working when starting. For example, failed GPU card and blank SSD are possible starting condition. So this sequence assume less working starting condition and is therefore more simple to succeed for most novice.
Macs with pre HS macOS first need to use full Apple upgrade process to get up to HS. HS installs CPU ROMs to handle APFS. Its been the case for awhile to go beyond HS, you first have to stop at HS. Don't know if OCLP folks included this. Best to be safe and install HS from Apple first.
More modern macOS definitely need to be on SSD. SSD installation will require solving for high fan speed due to lost of temperature sensor.
In general easiest with a pre-made Linux USB boot stick ( GRML )
1 out of 3 iMacs I tried couldn't use this method . Alternative method is much more challenging : A tool to clip onto GPU card's ROM chip and flash it. Thus, this step should be done at the very beginning to see if you are able to use the easier GPU flashing method.
Helpful to practice setting up and using this tool (can just read the current GPU info and its vBIOS and don't write it yet!) before removing the original working GPU.
Really important for AMD cards as the real flashing process will have a blank screen.
Practice will give you a sense of what happens and how long to boot up the tool before doing actual flashing.
SSD + 4GB (8 desirable) RAM is probably a must to run modern macOS. If still on HDD on these old iMac, probably want to do SSD and get a working system before anything else.
OCLP = Open Core Legacy Patcher. Its an open community of incredibly capable developers+testers+supporters building tools to keep older Macs running on newer macOSes. I guess Open = open source community, Legacy = Macs no longer supported by newer macOSes, Patcher = the driver+other patching requires to make these Macs work on newer macOSes.
If the Mac is still working, then learn how to OCLP upgrade it to higher (maybe even latest) macOS. This is a relatively big learning step. Best to know this before moving to +metal GPU upgrade which is another big learning step.
There are many combinations here. Even within a GPU card model, not all versions work! So need to search for actually what works. This website attempts to simplify as much as possible choose GPUs that work with the least amount of exceptions and incompatbilities.
Chance to screw up and brick boot is high. Could be bad used GPU, bad flash, bad install, bugs in software tools etc. It is actually possible to do +metal GPU upgrade and setup OCLP macOS while preserving user data and should probably only be done by the experienced. However, clean install seems to run the smoothest with patched OCLP macOS.
Opening iMac to install GPU have some damage risks. Will provide some pointers and tips. 2011 (at least 27") is significantly more difficult than 2009/2010 as GPU card removal only possible with entire logic board removed. Note the easy to knock off L5500 inductor on the 2011 27" iMacs (maybe 2009/2010 also? Not sure) link
Will be installing a blank SSD. This will force iMac to search for a bootable USB stick : First stick is GPU flash tool to flash GPU, 2nd stick is OCLP macOS installation.
Used GPU are sourced from PCs, need to be flashed with custom vBIOS to match how Macs boot
This is actually not too hard if
Using common and more matured +metal GPU card choices
Able to use the GPU flash tool in earlier step
OCLP has matured to the point this is just standard process even with upgraded metal GPU (maybe only true for the commonly used GPU for upgrades?)
Will want to run some basic graphics intensive app for 5-10min and compare GPU core and GPU heat sink temperature to check if heat sink installation is good.