My knowledge of what neighborhoods are currently great is a little rusty. I moved away from the area for a while, and now I'm one of those people with acreage outside of the city.
But when I was in the city full time, my favorite area was around 39th street just over the state line on the Missouri side. Walking distance to quite a few restaurants, coffeeshops, bookstores, and other funky shops.
Another fun place was the area around the Plaza. Some nice old apartments and houses. Walking distance from the Nelson art museum, the Plaza stores and restaurants, and fairly close to the bars in Westport. Westport is a fun place to hang out, but it's a little dodgy in terms of living there. Lots of petty crime, and a sad uptick in shootings of late. But when you're in your twenties, it's one of the most fun places in the city. At least it was for me.
One thing which changed after I left was the revitalization of downtown. Used to be dead, and now you can have a semblance of urban living there. Lots of lofts and townhouses available, and in walking distance of the new bar / restaurant district, plus the convention center and the new performing arts hall. Related is the River Market area. It's on the edge of downtown with lots of rennovated apartments & lofts. Has the main farmers market, and the usual bars, restaurants, and odd stores. And is very walkable. The new streetcars have a very limited route, but River Market and Downtown are covered (or will be).
The place I used to live now has a name: Crossroads Arts District. Stretches between downtown and the area around Crown Center. Has lots of art galleries, predictably enough, and all the other usual stuff. Holds a street art & food festival (First Fridays) once a month. Streetcar service will be coming soon-ish, but not quite there yet.
There's also the West Bottoms. Used to be a swamp, then was a warehouse / industrial district, then was abandoned (when I was there), and now is an up & coming area with the usual lofts and stuff. Used to be one of my favorite areas of the city due to the interesting abandoned factories and how quiet it was despite being in the middle of the city. Probably a lot of fun now.
I don't know much about the burbs. Johnson County (suburbia on the Kansas side) was great for big box stores but was otherwise a black hole for me. However, one great thing about Johnson County is that it has a Microcenter and it's glorious. Need a Raspberry Pi Zero + a handful of sensors for a project and need it now? Done.
Though speaking of the burbs, if you have a little extra money for buying, check out some of the inner ring suburbs. Places like Leawood. Some beautiful houses with old huge trees. Some of the more expensive neighborhoods in the city, but cheap compared to the Valley.
If you want to get off the beaten path, and are interested in fixer uppers, I always kinda liked Kansas City, Kansas. There are some neat old homes there on bluffs overlooking the Kansas River. And, last time I looked, many of those neighborhoods haven't gentrified, so there are bargains to be had. The neighborhoods are rough in spots, but there's something about the old blue collar neighborhoods that I always liked.
KCK and the area around Bonner Springs can also give you some country living on the edge of the city. There's still some farms left and still some acreage while being 10-15 minutes away from the city center. The only risk is that you buy your perfect quiet piece of land and then get swamped with development a few years later, but I suppose that's a risk for any city's edge.
Another good place for quiet small town-ish living while being very close to the city center is Parkville, MO. It's adjacent to the city. Probably one of the first KC suburbs. The town's center is Victorian and has a vibrant downtown with the typical Main Street USA feel. Also has a huge riverfront park on the Missouri River. The latter is quite nice because in much of KC, they've turned their back on the river and there's not much in terms of waterfront parks. This will be changing in the upcoming years with parks planned. But for now, there's not many.
One last place to mention is Lawrence, Kansas. It's about 30-40 minutes away from the city, and is the home for Kansas University. It's an absolutely wonderful place to live. Reminds me of Austin but without the crowds and high prices. Very vibrant downtown and music scene. Low crime. Very walkable for both the neighborhoods and in terms of hiking trails that start in the city. Tons of interesting old houses. Extremely laid back. Surprisingly hilly. One of my favorite places on earth.
Didn't mean to write a novel but that should give you some places to start looking. I spent much of my twenties in KC before heading out west in the dot-com boom, and then came back ten years later when I wanted some quiet times. I would sometimes be jealous of other larger cities, but KC was in many respects perfect for me. Unpretentious, unjudgemental (outside of the burbs, of course), and was a live and let live place from every angle. The biggest downside of KC is that a car is essential. I've tried to point out walkable neighborhoods, but public transportation is mostly limited to bus. No metro, no light rail (beyond a couple of miles of the streetcars). And a level of sprawl which is enormous. I suppose with Uber / Lyft these days, a personally owned car isn't quite so required now. But it was mandatory in my day.
On the plus side for cars, traffic is usually not too terrible outside of rush hour. And if you end up living in the city center and working in the suburbs, there's a good chance you'll be going opposite to the huge traffic flow, and even rush hour will be easier. That's what happened to me.
If you end up coming here, I think you'll like it. If I can help any further with your search, just let me know.