French press is always going to have some "mouth feel" because of the mesh filters.
I can highly, highly recommend the "Clever" dripper which James Hoffmann reviewed here. It's very similar to French press in method, great for single serve, almost totally foolproof, and uses paper filters for a very clean cup. I use Hoffmann's recipe and basically always have a good cup. There's a similar device called the Hario Switch which some people like more but I haven't tried it; the Clever has been great.
As for grinding your own, do it. Pre-ground coffee starts to go bad after a day, and just isn't as good. Start with an inexpensive burr grinder if you don't want to invest hundreds of dollars right away, but be aware that price/performance increases roughly linearly up to like $250, so spending more gets you a better result. Hand grinders like 1ZPresso can be even better for the price but are a little more work. The nice thing about immersion methods like the Clever though is that they're more forgiving of bad grinds, so you won't need to spend too much unless you want to use a drip method like V60.
You'll also need a cheap scale for weighing beans if you want consistent results. I recommend getting one that goes to 0.1 grams for single serve, with a large enough surface to hold your cup. It shouldn't cost more than about $20. I use this one.
You'll also want to experiment with beans. Modern coffee enthusiasts drift towards light to medium roasts instead of dark roasts, because you can extract more subtle interesting flavors and less bitterness. I have a penchant for very light, "natural" processed African coffees for their tart berry flavors. But it's all personal preference. There's way too much to get into in terms of processing, country of origin, blend vs single origin, etc. Try everything and see what you like. Coffee subscription services like Trade and Mistobox can be a convenient way to try a lot of things, or you can do what I do and just find a great local roaster to order from.
Btw, welcome! Check out James Hoffmann's channel on YouTube, it's my favorite coffee resource on the Internet. He has a ton of great advice in general. If you like print, he also has a small book called "How To Make The Best Coffee At Home" which is excellent and really all you need.
French press is always going to have some "mouth feel" because of the mesh filters.
I can highly, highly recommend the "Clever" dripper which James Hoffmann reviewed here. It's very similar to French press in method, great for single serve, almost totally foolproof, and uses paper filters for a very clean cup. I use Hoffmann's recipe and basically always have a good cup. There's a similar device called the Hario Switch which some people like more but I haven't tried it; the Clever has been great.
As for grinding your own, do it. Pre-ground coffee starts to go bad after a day, and just isn't as good. Start with an inexpensive burr grinder if you don't want to invest hundreds of dollars right away, but be aware that price/performance increases roughly linearly up to like $250, so spending more gets you a better result. Hand grinders like 1ZPresso can be even better for the price but are a little more work. The nice thing about immersion methods like the Clever though is that they're more forgiving of bad grinds, so you won't need to spend too much unless you want to use a drip method like V60.
You'll also need a cheap scale for weighing beans if you want consistent results. I recommend getting one that goes to 0.1 grams for single serve, with a large enough surface to hold your cup. It shouldn't cost more than about $20. I use this one.
You'll also want to experiment with beans. Modern coffee enthusiasts drift towards light to medium roasts instead of dark roasts, because you can extract more subtle interesting flavors and less bitterness. I have a penchant for very light, "natural" processed African coffees for their tart berry flavors. But it's all personal preference. There's way too much to get into in terms of processing, country of origin, blend vs single origin, etc. Try everything and see what you like. Coffee subscription services like Trade and Mistobox can be a convenient way to try a lot of things, or you can do what I do and just find a great local roaster to order from.
Btw, welcome! Check out James Hoffmann's channel on YouTube, it's my favorite coffee resource on the Internet. He has a ton of great advice in general. If you like print, he also has a small book called "How To Make The Best Coffee At Home" which is excellent and really all you need.