What's the best French Press? Is there a particular brand that's really good?
How much should a good one cost?
Bodum. Buy a Bodum.
If you don't like much sediment, buy a nylon filter to add to the stack. They're available OEM. Sweet Maria's usually has them, but sometimes run out.
The English call a French press a cafetiere. Coffee geeks call them FPs. Call them what you will they make a civilized but dynamic cup of coffee.
FPs do well with darker roasts, even roasts as dark as "Viennese" and "French." If you like mediums, you may want to consider some sort of pour-over instead or in addition.
FPs are available with insulated pots. If it takes you more than twenty minutes to go through a pot, you may want one of those. No way you're going to get more than 45 minutes, though. Unlike espresso, FP coffee is sturdy enough to reheat fairly well in the nuke; but don't let it boil.
When you're wondering "what size?" Remember, an FP "cup" is only 3 oz. An "8 cup" Bodum is two mugs, just big enough to get your heart started and the rest of you out the door on a weekday morning. It's not big enough for two people.
Even though an FP uses a coarse grind, it will benefit from pairing with a good burr grinder. Good enough for an FP means something inexpensive, but in excess of $100. You don't need an espresso level grinder. You can make an FP work OK with a "propeller" blade grinder; but it's a definite step down.
FPs are very revealing. If you use coffee ground more than a few minutes before brewing, it will not be as good as fresh ground. The fresher, the better when it comes to grinding. An FP will also reveal the faults of stale or bad coffees.
Cold brew is seductively mellow; but not very revealing. Great brewing method for people who aren't sure really sure if they like coffee or not, the in-the-cup results are similar to vacuum pot brewing's. If you're after excitement, cold brew might not be your best choice.
Of course it's great for iced coffee, is the only way to go for "iced blendeds," and is good for all sorts of cooking, e.g., sauces, ice creams, tiramisu, etc.
The downside, such as it is, is that you have to plan ahead. No instant gratification until you get the essence in the fridge.
As always, you want to use good coffee and a good grinder.