I did the dho dham tour a few years back during June. I found Kedarnath to be really a wonderful trekking experience. I went with my parents and some other family members. They hired ponies for the trip, but they really complained of saddle butt afterwards
I decided to walk up, and it was not too bad. It is especially nice to be able to take breaks, photos, etc. You can have a quick snack about halfway up. I was walking pretty swiftly despite that, so I reached Kedarnath about 60-90 minutes after the ponies did. There were lots of people, most of them making the journey by walk. If you are able, I really think it is the most comfortable way to go, as the ponies and sedan chairs can be pretty uncomfortable if you are not used to them, especially after a few hours.
The only downside, if you want to call it that, is that June is a pretty rainy time of the year up there. However, when I was there, the rain was not particularly heavy, and during the hike, it really helped to cool the temperature and I didn't feel as exhausted as I thought I would.
The trek back down the next day was very easy by foot... i reached the bottom about half hour before the ponies did. It was sunny that day, and I got to take some snaps of things I had missed on the way up.

Badrinath was also very pretty, with the river flowing just in front of the temple. The lighting at night from across the river adds a nice touch. I did not go to the bath houses, but it might be something to consider...
When you are up at Badrinath, try to make the rest of the trip up to Mana, the last village before the China border. Some really interesting places here, like the supposed origin of the Saraswati River and the place where Vyasa is believed to have written the Mahabharata.

We did not visit Gangotri or Yamunotri, but I can only assume they are equally appealing in their own fashion. Personally, though, I do not feel as though I missed much by not visiting those two. From everything I have heard from others who have done the trip, those two sites are also relatively difficult to access and the last portions of the trip must again be made by foot or pony. For us, it was a nice experience to make the trek up to Kedarnath. Having to make the same trek two more times, each even longer than the trek up to Kedarnath, would not have gone too well with the other members of my group, who had already been quite... hobbled... by the pony ride up to Kedarnath

Have a nice trip! I am sure it will be enjoyable whichever season you choose to go.