Another coffee fanatic checking in. Everyone has great points. I'll just throw mine out there:
First -french presses will leave a dust/mud (called "fines") in your coffee. Unless you have a good burr grinder, then put it in a sifter to get the dust out, you'll still get it. French press does make a nice pot of coffee though. I recommend a stainless one, look for something that labels it as made from 18-8 (good stainless, higher number better), from a not sketchy manufacturer. If you buy a cheap one, you will taste the metal, since coffee is highly acidic it'll leach the steel. French presses can't use pre-ground very well, since pre-ground is dust heavy.
Second - pour over/dripper coffee (not drip coffee), it has a simple plastic/glass cone that sits on the mug, or bigger ones for pots, then a paper filter inside, is best. It'll provide a clean, dust/fines/sediment free, and pull the taste just as well. Also it's able to make a very strong coffee. They're also like $5 for the cone, and $5 for a pack of 200 filters.
Other coffee options: aeropress (quite nice), "Chemex" (fad, it's just pour over), percolator (nice), coffee machine (easy), espresso machine (great but involved)
Alright, essentially, all methods of making coffee are pretty good, some need more work, some are cleaner, but the big point is, if you're using crappy beans, and adding flavourings, it doesn't matter how you make it. You need to locate good coffee! This is the biggest component.
You can google your area for "coffee roasters", "third wave coffee", "single origin coffee", and any other hipster terms + coffee. You need to locate someone who gets beans from a specific farm, and roasts them properly. They'll sell the coffee with the cupping notes (cupping is the tasting of the coffee that is done at the farmers market in the coffee producing country, which is followed by bidding for the lot of beans). Notes will be "chocolate, stonefruit, cherry, blueberry, agave, cane sugar, bright, clean, complex, melon" etc. This stuff, it's like rib-eye vs round or cheesewiz vs. 2 year old hickory smoked Gouda, or whatever else. Something to mix into your diet occasionally.
Next bean source: there is amazing coffee all over the world. "African coffee is acidic" is like saying all "Asian people are smart". That's a pretty broad brush. Stuff from Nicaragua is vastly different from Sumatra or Ethiopia. My personal favourite is Ethiopian. Even the elevation within the country and the soil changes it. This stuff is like grapes.
To find economic coffee, you'll have to essentially just try all the ones in the grocery store. I tried about a dozen, before I found a pre-ground that was decently good. Don't bother with anything Starbucks or McDonald etc, or any of the commons advertised on TV, big waste of time. Try the slightly different, but stocked in good quantities at the grocery store.