Tokyo - Japan's capital (DUH!), where almost all of the places in the anime were located like Tategami, Shinjuku, Tama River, Tokyo Terminal station, and many more.

Tategami School - where the AEGIS base hid underground. Tategami was the official school of GateKeepers for easy access to the base. There were several hidden passages in Tategami that led to the base. I'll give you some that I noticed: trash bin (for Shun and Ruriko), a cubicle in the girl's restroom (Reiko), underneath the mats in the gym (Kaoru), garden (Bancho), a teacher's table? (Megumi), Science Library, Principal's chair (Shirei), men's restroom (Bancho), underneath the volleyball holder (Kaoru), locker room (Shun), fire extinguisher cabinet or a fuse box? (Ruriko), and the one for Fei - I don't know how it's called but it is like a cup shaped thingie ^^;.

Futago Tamagawa - not a setting in the anime, but it was mentioned in episode 1. Saemi recalled to Shun that once they rode a train that is headed to Futago Tamagawa with their father.

Shinjuku - youths were having a concert in this place when the Invaders attacked and trapped every one of them. To be specific the electrical wave was detected near West Entrance of Shinjuku.

Town Hall - not specified where the Town Hall is, but it was where Reiko performed her recital for charity.

Nagoya - Reiko, Ruirko and Shun were aboard the Shinkansen (bullet train) to investigate a GateKeeper candidate (who happens to be Bancho), the train stopped at Nagoya, where Reiko gets off to buy lunch. She was left behind when the train heads further.
Info: a port and city in central Japan, capital of Aichi prefecture in the Chubu region of Honshu Island. It is the 4th largest city in Japan. Landmarks include Nagoya Castle and the Atsuta Shrine. It has iron and steel works, and aircraft, automobile, and chemical industries.

Himeji castle - where Ruriko and Shun found Bancho fighting with a group of bandits.
the Himeji Castle, otherwise called "Shirasagijo" (White Heron Castle), is situated in Himeji City of Hyogo prefecture about 50 km west of Osaka. It is about 370 years since Himeji Castle was constructed in its present shape. This is the only excellent castle which architecturally represents Japanese culture and, at the same time, has handed down its original design at the time of construction.
A white castle tower looms high in the sky. Around it, numerous turrets and plaster walls are arranged. Stone walls describe elegant curves. This symmetrically beautiful silhouette reproduces the graceful figure of a heron about to take wing at any moment. From: Himeji Castle - The Architectural Glory of Japanese Castles.

Matsumoto castle -

Nijou Castle -

Goryokaku -

Fuchu Base - not sure about this, but in ep. 4, Shirei asked Miss Ochiai if "they" have already arrived from Fuchu Base. I think "they" refers to the cargo planes and Gate Engine, so I assume that Fuchu Base is where cargo planes are hidden.

Hokkaido - mentioned location, where Reiko (ep. 5) wanted to go because of good food.
Info: the most northerly and the second largest of the four main islands of Japan, constituting an administrative region. Separated from Honshu Island by the Tsugaru Strait but linked by the Seikan Tunnel, its chief cities are Sapporo (reg. capital) and Hakodate.

Kanagawa School - the school where Kaoru came from.

Metro Highway - where AEGIS scouting helicopters searched for the Unidentified Ground Object. Invaders destroyed this highway, so AEGIS searched within this area, when the fact was Invaders were hiding themselves in Toma Highway.

Toma Highway - where the Invaders hid themselves in forms of road plates. It was a newly constructed highway, but the Invaders destroyed it after tricking AEGIS.

Kaminarimon Gate - where the GateKeepers fought the huge Invader Lantern poorly, leading to Shirei's decision of choosing a captain.

Violet Dormitory - a dormitory for girls in Tategami School, where the "Forbidden Rm. 104" was located. Kaoru and Reiko lived in this dorm when they transferred to Tategami.

Haneda Airport - the very well known airport in Japan

Atsugi Airforce Base -

Kawasaki Oil Refinery - where the GateKeepers first met Invader Executive Shadow. Shadow chose this place probably with a plan of burning the GateKeepers "like bugs".
Info: (Kawasaki) a major industrial city on the southeast coast of Honshu, situated northwest of Tokyo. It is the capital of Kanagawa prefecture in Kanto Region and has steel mills, oil refineries, electrical, and engineering works.

Bermuda Triangle - where the cargo ship, Marine Sulfur Queen disappeared.
Info: an area of the western Atlantic between Bermuda and Florida, with a number of unexplained disappearances of ships and aircraft.

Shin Osaka - another mentioned location, it's where the Shinkansen, Hikari came from (ep. 11). Maybe a "Shin" (New) is added to the word, because it was rebuilt after the damage it received during the World War II.
Info: a port, commercial, and cultural city in central Japan, on Honshu Island, chief city of the Kinki Region. It is the 3rd largest city in Japan and lies on the northeast shore of Osaka Bay. A great comercial center during the Edo period, the city was largely destroyed during the WWII, and later rebuilt. It is a center of transport with ferry links to Kyushu and Shikoku with many light industries including tourism.

Kyoto - aboard the Shinkansen with Saemi, Shun pointed to her the Kyoto Tower while passing.
Info: an industrial city in the Kinki Region of central Honshu Island. Founded in the 6th c., it was capital of Japan from 794 - 1868. Historical landmarks includes the 14th c. Golden Pavilion, 15th c. Ryoanji Temple of the Zen sect, and the 17th c. Nijou Castle. Undamaged during the WWII, it has many industries including electrical equipment, precision tools, and cameras, as well as food processing, chemicals, and textile machinery.

Kobe - mentioned location, a place Shun wanted to visit if ever he would win a lottery.
Info: a seaport in Hyogo prefecture, central Honshu Island on Osaka Bay. Since it was built in 1868 its harbor has become one of the leading ports of Japan. It has steel and shipbuilding industries. It was devastated by a major earthquake in 1995.

Kyushu - mentioned location, an island Bancho (ep. 5) and Saemi (ep. 11) want to visit.
Info: the most southerly of the four main islands of Japan; constituting an administrative region. Chief city, Fukuoka.

Tama River - location, Shirei chose to intercept the Shinkansen Robot because of the train's speed and geographic features of the place.

Mt. Daisetsu - the cold mountain where Shun and Ruriko found Yukino. The first time Shun and Ruriko visited this place, it was filled with Iwatsumesou flowers. Mt. Daisetsu was actually a snowy mountain where Yukino far away from the population of Japan. (credit to ShunUkiya for reminding me of this place)

Dream Island - an island surrounded by the Tokyo Bay. It was where Kikai Shogun and Akuma Hakushaku met their defeat.

Yoyogi Park - place of assembly of the Society for the Study of Political Thought, headed by Reiji Kageyama.

Ueno Station - train station where Miss Ochiai's Invader fiance took her instead of the limited express.

WHK Hall -

Diet Building -