Well, what I'd probably do is move out somewhere in the Lenexa / Shawnee area. It's a nice quick drive down to work (and you'd be moving opposite the big bulk of traffic, which works downtown and lives in Olathe). Reason I'd do that is because of Shawnee Mission Park, a huge park with a fabulous off-leash dog area. It would be really convenient. Also, if you're close to I-35 and/or Shawnee Mission Parkway, it's a very quick jump to the 20-something nightlife of Westport or P&L. Only downside is it's pretty tame and suburban. You'll be driving for your getting-out time, but you'd be close to work and there's lots of apartment choices of every type.
Placing more priority on living close to work...don't. Olathe is not exactly a happening 20-something scene. It's very...suburban family. Not that that's bad, but it doesn't sound like you.
Downtown, there are some new condos in what's called the Crossroads area, centered at 20th & Baltimore if you're looking at a map. Very artsy, hipster area - lots of fun; we looked at living down there, too. Also, the River Market area, up around 3rd & Walnut has some new apartments and condos, new renovation - very nice.
Anywhere in the downtown loop (the rectangle of freeways) is nice...not sure how much residential there is there, though, mostly commercial.
Around the Plaza and Westport area there are a lot of places, too, can't go wrong with those, though the Plaza area is quite expensive. I'd happily live in the strip between southwest trafficway and Main St, anywhere up and down that.
You might also look at the Waldo neighborhood, 75th & Wornall on the Missouri side. It isn't downtown, but it's got a nice little urban vibe, very cozy. Similar neighborhood would be Brookside, 63rd & Brookside Blvd - also a lot of fun shops, good community spirit, somewhere I'd recommend newcomers look. Lots of good places to take your dog in both. If the bistate tax issue (see below) doesn't bother you too much (and being single, it probably shouldn't), these would be my top choices. We would have moved here if it weren't SO far from work.
Oh, and I'd be remiss if I didn't mention West 39th Avenue, on the Kansas side, 39th between Rainbow and State Line. Also an up-and-coming hip sort of area, with lots of independent restaurants and shops. I've got a friend who lives there with her dog, and I don't know about a park, but there's good walking opportunities at least.
Rougher areas...Jackson (MO) and Wyandotte (KS) counties west of I-35 or Jackson County (MO) east of about US-71/Troost Ave. can get pretty poor and pretty rough. You'll want to avoid that.
You should be aware that since we live in a bistate area, people who live in one state and work in the other pay some income taxes to both states. Also, KCMO takes a cut of your income if you live OR work in the city. Might be something to think about.
Oh, and the other thing I'd think about is that generally, moving north-south is very easy in this city, moving east-west less so. So you can plan for that.