There are many great places in the Pokemon games. I want this to be fair so I'll list some favorites from each region.
Kanto: I like the Pokemon Mansion. It is an abandoned, partly destroyed building with scientists roaming around freely and doing weird experiments. There are also hints to legendary Pokemon in the diaries, adding some extra lore to the games. Plus a puzzle featuring hidden switches in statues, and a reward at the very end of the building. Cinnabar Island itself is pretty cool too, I like how it is an Island rather than just a regular town. Speaking of Islands, I am also very fond of the Sevii Islands, they feel very unique and varied with a lot of interesting and cool environments.
Johto: As others have mentioned, Ecruteak City, I think it captures the traditional Japanese feeling better than anything else in Johto. I also like the Bellchime Trail with its colorful autumn leaves, plus Bell Tower itself which is a really cool tower, standing at the top and meeting Ho-Oh is one of my favorite moments in the Johto games.
Hoenn: The Abandoned Ship in R/S/E, and Sea Mauville in OR/AS,
Max. Optimizer explaned it better than I ever could. A really cool place with lots of mysteries, very fun to explore. I also really like the Mirage Spots in OR/AS. While they were quite small and didn't have a lot of content, I think their individual designs are just fantastic. They all look unique, several of them has some sort of unique feature such as a volcano, ruins or a meteor crater. I am also a huge fan of the Battle Frontier in Emerald and the Battle Resort in OR/AS. Not just for their features, I also like how both areas are designed.
Sinnoh: This region has many great places, but the first ones that come to mind for me are Snowpoint and the surrounding routes. I love snow and seeing it in a Pokemon game for the first time was amazing. I am also a big fan of Mt. Coronet, climbing it on the outside and not just on the inside made it feel like an actual mountain as opposed to just another generic cave. It was long and complicated, I know many dislike it just because of that but I liked it a lot.
Unova: This is my favorite region, so there are a lot of places I love. Can I say all of it? No? Then I'll try to be a bit more specific, going to mention some of my top favorites.
First is Aspertia City. To me, this is the best starting city in the series. It has a Gym, a Pokemon Center, concrete paths and several houses apart from the player's. I also love the lookout, standing there and watching Unova is fantastic. It gives a feeling of building something on the past while looking towards the future.
Then there's the Strange House. A creepy place done just right, the fact that the furniture moves on its own is just one small detail, but the atmosphere it creates is incredible.
Black City, and Black Tower. For me, they manage to create the feeling of a "big city" better than many other places in the series. It is hard to explain, it just gives me the feeling of being in a big city more than anything else. It might have to do with the darkness, or the black coloration of most buildings, I'm not sure. Though Castelia comes very close, as well as Lumiose but in a completely different way.
Celestial Tower. A typical burial ground, but without being creepy. Instead, it is a place which shows respect for the departed. Some other places in the series has similar areas, notably Lavender Town in the Johto games, but I feel that the Celestial Tower does it the very best. That's not to say I dislike the creepy places in the series (as seen above with the Strange House, I also like Mt. Pyre and the Old Chateau), but not every area with ghosts needs to be creepy to be well done. The Celestial Tower shows that.
Kalos: Route 8/Muraille Coast. I really liked how this was split in two, one upper section on the cliff and then a lower section on the beach. The view from when you first enter from the Connecting Cave is really great, I loved seeing Cyllage City below, wondering what place it was and when I would get there.
Alola: I'm going to have to say Aether Paradise. The upper floors give a sense of what they are actually supposed to do, protecting wild Pokemon and keeping them safe, people doing honest work to help Pokemon and the environment. Then there's the basement which is completely different, it gives off a creepy wibe similar to when you first discover the secret laboratory in a Metroid game, the first instinct you get is that there's something that's not quite right there.
As for the side-games... I have only played XD and Colosseum (as well as PBR, but it doesn't really have any "areas"). I think they deserve a mention since they have some wicked awesome places too. I really like The Under, an underground city is really cool and this was well done. The Relic Stone at Agate Village has an atmosphere of safeness and relaxation, I always liked coming there to purify Shadow Pokemon. Gateon Port in XD was cool, it felt like a real harbor more than any other place in the series (at that point). Citadark Isle was amazing as well, an area with lots of variation but also very dark, a perfect place for the final showdown against the evil team.
I might write about more areas later on, but this is it for now.
