When I think of "classy restaurants", I think of snobby restaurants, which is not really my thing. Fancy dress, nosebleed prices, and food I probably wouldn't choose to eat, much less pay a hundred bucks for. No thanks. I'd rather eat at "good" or "interesting" restaurants than "classy" ones.
Around here, I like Clara's. Great little Italian restaurant, good place to take a date but also fine for a family jeans and t-shirt meal. Maggiano's Little Italy is good as well. My city's downtown area has a ton of decent or interesting restaurants, not sure you could ever hit them all.
I went to college in southern New Mexico and had some favorites there as well. Chope's is in La Mesa, a tiny town of maybe 100 which is actually on the other side of the Rio Grande but still in the US. This is basically somebody's house which has been converted into a restaurant. You sit in what was probably a living room at one time and can see a normal home kitchen with a larger industrial kitchen behind that. There is a nice painting of the owner's wife (I guess Chope died a couple years back) hanging on the wall and you sometimes see her in the kitchen cooking. Amazing New Mexican food, basically a home-cooked meal. Somehow everybody knows about this place even though it's really out of the way.
Si Senor is a chain of about 3 restaurants and is a little nicer and more commercial. It's where you take the folks when they come into town. It's an adobe style building with dozens of palms outside. Inside, the center of the dining area has a large water fountain and stage area where you'll almost always find a live mariachi band performing over the dinner hour. The burrito and enchilada platters are amazing with red or green chile. They also serve chips with a few sauces before the meal - a cold red chile salsa, a warm green chile sauce, a cold green chile and sour cream dip, and a spicy bean dip. There is a seperate bar area which is pretty small but has a gorgeous starry-sky at twilight mural on the ceiling, almost enough to fool you into thinking you're sitting outside.
Mesilla Valley Kitchen is a breakfast/brunch place open from about 5 or 6am till 2pm. The menu is painted large on the wall where you come in and is very diverse. They've got traditional breakfasts like bacon and eggs or pancakes as well as regional food like huevos rancheros and breakfast burritos. Aside from that, they've got several sandwiches (served with hand cut and fried potato chips!), burgers, salads, and a couple other New Mexican dishes. My favorite breakfast there was "spuds", a skillet with diced fried potatoes, bacon, sausage, chorizo, melted cheese, and red or green chile sauce. You could also order one or two eggs on top and it was served with a fresh tortilla. Good stuff! The place has kind of a relaxing homey feel with typical white and turquoise southwestern color scheme and some nice paintings of the local mountain range, chile ristras, and so on.
We didn't have a Starbuck's, so I'd sometimes go to Milagro Coffee. They had a huge number of coffee varieties and rotated them often so you'd go in and choose something like light/regular/dark/decaf/flavored, but what each of those was usually was different from visit to visit. They had espresso drinks of course, but I usually stuck to the plain coffees (dark please!). Comfy furniture and a lot of students hanging out or doing homework. Coffee "for here" was served in a mug or large cappuccino cup, which was nice. If I was up early, I liked to get mine to go and sit outside on a brick wall - it was a great spot to watch the sun come up over the mountains.
None of those places were really expensive or fancy, but they had class as far as I was concerned, or at least a certain charm you don't find at Chili's.
