Downtown and Kyoto Station:
Kyoto Station is the one of the hubs of the public transportation and is attached with many shops, but not in the heart of the city. The blocks surrounded by Kamo River, Karasuma-dori(street), Shijo-dori, and Oike-dori are the heart of the city. You can reach there from Kyoto Station by subway easily. Downtown map is here.
Dining:
The following page is useful to find places to dine.
http://www.kyoto.travel/dining/
One of the recommended streets to dine is Pontocho between Kamo River and Kawaramachi-dori: you can find many small restaurants on the beautiful alley.
TIPS when arriving at Kyoto Station by Shinkansen:
At first, go down to the concourse from the platform by any stairs or escalator.
If you want to transfer to the subway, Shinkansen Hachijo East Exit (ticket gate) is convenient. Follow the sign "Transfer to Subway".
If your want to take a taxi to your hotel in the downtown (the north side of the station), exit at Shinkansen Central Exit (ticket gate) and turn to the right to the north side (Karasuma Side) of the station.
If your hotel is on the south side of the station, exit at Shinkansen Hachijo Exit (ticket gate).
Tourist Information Office at Kyoto Station:
When arriving at Kyoto Station, we suggest you to stop at the main tourist information office of Kyoto (Kyo-Navi). It opens from 8:30 to 19:00.
If you arrive by Haruka from Kansai International Airport, exit at Central Gate (ticket gate), turn to the left, go up to the second floor, and turn to the left. It is on the way to the south side of the station.
If you arrive by Shinkansen, exit at Shinkansen Central Exit (ticket gate) and turn to the right. It is on the way to the north side (Karasuma Side) of the station.