This is for the cold valve, which I need to remove to replace. But do not know the model or how to remove it.
You've done well so far. Most people have trouble getting the handles off. Make sure you have your shutoff valve under the sink turned off. If you don't have one, you'll have to turn off the water to the house. Now get a large adjustable wrench and adjust it tight to the brass nut around the valve stem and turn it counterclockwise to remove the valve stem. There might just be a washer on the other end of the stem that can be replaced. If not, bring the stem to your home store or plumbing supply store and they should be able to get you a match. I can't tell from the picture what the model faucet this is but it's very similar to an discontinued American Standard model that has a cartridge looking like the one below, from Ferguson plumbing for reference. https://www.ferguson.com/product/american-standard-ceramic-disc-hot-and-cold-cartridge-for-moments-2506921-and-2506920-a9940530070a/_/R-1650277?fromplp=true
It's a ceramic cartridge so it might just need some cleaning out of particles.
Insert a screwdriver here and pull that retaining clip out.
Now you can remove the whole upper part, that should expose the valve body and the nut that is holding it.
Meanwhile you can remove the decorative ring, it is not held by anything.
Use proper wrench to unscrew the valve (assuming you turned off the water first.
There are two seals inside. The top one is for blocking the valve leaking to outside world. The bottom one is for valve opening and closing water flow. If noting else is wrong, you can just replace those two seals.