Horyuji is the Buddhist temple located in Nara prefecture (Kansai Region), and has been registered as the first World Heritage Site in Japan. It is said to be the oldest surviving wooden structures in the world, which was founded by Prince Shotoku and Empress Suico in 607.
Prince Shotoku is very famous in Japan for learning the advanced culture of the Chinese continent by dispatching the envoys to Sui Dynasty China, and then creating the centralized government and spreading Buddhism in Japan.

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Recommendations

Horyuji is the most important temple in this area, but Hokiji nearby is also registered as a World Heritage Site. Many tourists have a small trip to Hokiji after visiting Horyuji temple.

Horyuji
This Buddhist temple is the oldest surviving wooden structures in the world which was registered as the first World Heritage Site in Japan. It consists of the two temples, West Temple and East Temple, and there are many structures in each. I think you should see Center Gate “Tyumon”, Main Hall “Kondo” and Five Storied Pagoda in West Temple, and Main Hall “Yumedono” in East Temple at least.

Small Trip from Horyuji to Hokiji 
Hokiji is the temple founded by the son of Prince Shotoku. Three Storied Pagoda here still retains the original features, and this temple was also registered as the World Heritage Site. On the way to Hokiji, you will find other temples such as Tyuguji and Horinji.

Access

The nearby railway station is JR (Japan Railway) Horyuji, a little far from the center of Nara town. You can walk to the temple in 20 minutes, but I would like to recommend a rental bicycle so that you can enjoy other temples such as Hokiji after vising Horyuji. You can take it at the shop near the station south exit.
You should refer to this site to see the public transportation to JR Horyuji Station. It takes about 1 hour from Kyoto, 30 minutes from Osaka, and 10 minutes from Nara Station.

Travel Route

Both Nara and Kyoto are famous for their traditional townscapes, but I think there is a big difference between them. Nara has more quiet and relaxed atmosphere than Kyoto. That is why I would like to recommend you stay in Nara for a while.
If you stay near the Nara Station, you can enjoy a day trip to Horyuji, and then walk around Nara town to enjoy the old temples such as “Todaiji” which is very famous for the great statue of Buddha, the park “Nara-koen”, the preserved traditional town “Nara-machi” and so on.

 Day Move Sightseeing Stay
1 Kyoto/Osaka -> Horyuji -> Nara (Japan Railway) Horyuji
Hokiji
Nara
2 around Nara (walk) Todaiji
Nara-koen
Nara-machi
Nara
3 Nara -> Kyoto/Osaka (Japan Railway)
Kyoto/Osaka -> Next (train/bus/plane)