The "classic taste of Japan" for first-timers usually includes some time in Tokyo and surrounding area and some time in the Kansai region - Kyoto, Osaka, Nara and surrounding area.

For 7 or 8 days in Japan, I'd suggest 3-4 days in Tokyo and 4-5 days in Kansai. Since Kyoto, Osaka and Nara are all within a short train trip from each other, you really don't need to change hotels. Just find a good place in Tokyo and one in Kyoto (my preference) or Osaka.

Tokyo is the quintessential ultra-modern Japan and Kyoto has the old, traditional castle and temple vibe. Nara is even older than Kyoto and was the first permanent capitol in the 8th century.

The best way to do all the above is to arrive in Japan at NRT (the New Tokyo International) or HND (Haneda-Tokyo International) airport and depart from KIX (Kansai International Airport) outside of Osaka - or vice versa. If you can't arrive/depart to/from different airports, then select one, probably Tokyo. In that case, you will need to plan a round trip on the bullet train (shinkansen) to get to and from Tokyo and Kyoto.