Since you mention wanting to filter really well, I would recommend you take a look at chemex. It’s a pour over type of brewer but their smaller model is for a single serving and will be more lenient with the pour method since the chemex filters are so dense. Other pour over brewers will require more of a setup to maintain consistency (but you can also tweak the brew to your exact liking in this regard!).
Water temp and grind size are somewhat critical for quality coffee but you don’t need it right off the bat of course. I would get a grinder before a kettle, tbh; fresh grounds make a bigger difference than a few degrees difference in water temp. Do work on “calibrating” your water temp in whatever method you use, though. If you microwave the water, figure out how long to let it cool before it reaches 200 or whichever temp you find brews best for your beans.
Search for local roasters and buy some beans from them. Keep searching until you find a brand you like. Grocery store beans are very mid. (There are many weekly “drop” style bean shippers that are pricey but let you discover online roasters that will ship you beans; it’s been invaluable for me to find really good roasts!)
As for cold brew, it’s a separate beast entirely and I wouldn’t explore it just yet. I used to make large batches for my nitro setup and did that in a bucket. Definitely super smooth but a bit more work. You can make smaller batches with a French press. I’ve always gotten my presses from IKEA because they’re cheap yet effective. Far from fancy. I would never consider heating up cold brew personally.