I've been on the bullet train and there is plenty of room or bags, they run every 30 minutes so they don't tend to get too busy.

From the Kyoto station to city center it's about 10 by taxi.

Don't forget to get a PASSMO subway pass they are the easiest way around, a taxi will set you back 700 Yen for the 1st 2KM.

In Kyoto, visiting the Nishiki Market is fun, you also want to check out the Gion district, and the Nijo Castle.

Tripadvisor has a lot of ideas ready to go.