I've been in the coffee business for over 25 years. There's as many ways to make coffee as there are coffees! What all of the 3rd wave brewing methods have in common are the satiety provided. You are extracting as much of the good qualities out of the coffee as possible without extracting the bad.
Probably my favorite is Hario
HARIO Co., Ltd. -
Makes the 2nd smoothest cup in my opinion and is what you will see in most 3rd wave coffee shops. Clean up is simple. Once you get an eye for the measurements you can adjust to your taste.
French press does a nice job, but makes a "muddy" cup in my opinion. I like the flavor that you get with a FP, but you have to be ready for the grittiness. Clean up is a pain if you are water restricted.
Close to the Hario in being my favorite is the Toddy cold brew method
Cold Brewed. Simply Better. | Toddy Cold Brew Coffee While the home system can be a little tricky to use, I have seen a few new systems introduced but have not tested them yet. Clean up is messy, but you can make the concentrate at home and take it with you. Cold Brew coffee is the fastest growing segment in the beverage category. The absolute smoothest cup of coffee possible. It's about 67% less acidic than hot brewed coffee, so better for your teeth and stomach. Be prepared to taste what the coffee really tastes like! It's not just for drinking cold either. Because you are making a concentrate, you can add boiling water to it (how I do it) and have a ready to drink cup, or add cold water and heat it up.
There are a lot more and I could talk for hours on coffees and brewing systems, but the average "Joe" is pretty happy with the good old home brewing system. If you want to get really nerdy, check out the Specialty Coffee Association of America page on approved home brewers. The one listed first, Technivorm Moccamaster is a machine I have had since 1995. I've never had to work on it and it's been on the list since its inception.
By the way, I am not associated with any of the companies products I recommended above.