A few photos of HIMEJI castle. Well worth a visit.

Just to note that its 15 min by Shinkansen from ShinOsaka station to Kyoto. Taking other trains from other Osaka stations and trains can take 30-45 minutes.

That is a good point if you are staying next to Shin Osaka and had a rail pass. Shin Osaka would be a good base to visit Kyoto, Himeji, Hiroshima day trip, etc, but would also require commuting to Osaka to see the sights there.

Most visitors to Osaka stay closer to the main rail stations rather than to Shin Osaka. Two stations near each other: JR Osaka Station and Hankyu Osaka-Umeda Station that can be used to get to Kyoto.The Hankyu (non-JR) train has a couple of useful stops in Kyoto city and might be better than JR depending on where you are going in Kyoto (and to some extent, where you are starting in Osaka).

From JR Osaka Station to Kyoto Station:
Time required: 0:29 Price: 580 YEN on JR Rapid Service train
Time required: 0:29 Price: 2,070 YEN on JR Limited Express
Time required: 0:29 Price: 3,280 YEN using Nozomi shinkansen

The rapid service is frequent enough that you just show up at the station and take the next train (which probably originates there and empty when boarding). That costs all of $4.

The limited express is a nicer train but less frequent and higher cost if no pass. Am pretty sure it stops at Shin Osaka.

The shinkansen option would mean taking the rapid service train 5 min to Shin Osaka and 10 min to transfer to the shinkansen for the 15 min ride to Kyoto.The total travel time could be 28 to 31 minutes. You could save 200 yen, I think, if you didn't take Nozomi, but that would reduce your options significantly. But, that might be your only shinkansen option if your pass doesn't include Nozomi (some passes exclude all shinkansen between Shin Osaka and Kyoto)

That rapid service train you take from Osaka to Shin Osaka is likely going on to Kyoto. Your choice is then to stay on that train or spend your time transferring to the shinkansen (much more comfortable, but Shin Osaka is big). Either way, you get to Kyoto at about the same time.

Don't ask me why I went into all that. Bottom line is that it is not 15 min from "Osaka" to Kyoto by train. It is double that or longer, depending on where you are starting the trip. Osaka is a big city.

Thank you everybody for helping planning the trip to Japan!
InsideOsaka.com and InsideKyoto.com turned out to be great sites with so much helpful info. We are back and I am happy to report that that segment of the trip before going to Korea for 13 day was excellent.
We ended up getting a 7-day JR pass which we pre-ordered in US prior to price increase and picked up in Kyoto. When we picked up the pass, we also got tickets to Osaka, round trip to Hiroshima. Some local trips with JR pass like to Nara, Arashiyama did not require tickets.
After we flew to Osaka we took a pre-ordered taxi (about $130) to the Cross Hotel in Kyoto. Since we arrived pretty late, we figured out that taking cab will save us time since we did not have to stand in long line to get the pass and figure out trains and transfers to Kyoto. We stayed in Kyoto for 3 nights and saw most of the sights we planned. Kyoto is definitely an open air museum. Then we stayed in Nikko Hotel in Osaka for 4 nights. Used JR to visit Nara, a pretty full and great day, and Hiroshima with its museums. It was nice also to use JR ferry to also visit Miyajima.
Happy travels!