Depends on your character, but I choose to remain neutral, although I am a STRONG Stormcloak supporter.
I can't bring myself to kill Whiterun Guards, nor can I stand to watch Balgruuf get beaten down and surrender. It makes me feel uncomfortable, plus I dislike Vignar.
It will also destroy parts of Whiterun and maybe Solitude, and they will not get repaired.
It will not affect your Housecarls. Even Farengar remains Court Wizard of Whiterun for Vignar.
To give you some insight, Ulfric was training to be a Greybeard, that's a pretty high honor. He then left High Hrothgar to join in on the Great War, which he fought in and was eventually captured. The Thalmor twisted his mind and would ultimately cause the rebellion. After being freed and after the war ended, Ulfric led a group of soldiers into Markarth, using the Voice to shout open the gates, and retook the city from Forsworn control (who had conquered it 2 years prior while the Empire was concerned with the Great War). He became Jarl of Windhelm and eventually challenged High King Toryyg to a duel, shouted him down with his voice, and pierced his heart. This threw Skyrim into an uproar and divided it, thus beginning the rebellion.
Some say Ulfric is out for power, others say he's true to his cause. I think he IS true to his cause. He was important enough to become a Greybeard, he also fought for the Empire, retook Markarth, and challenged Toryyg fairly. He even surrendered so his men would not die in vain.
EDIT: Many Nords were pissed off because so many of them died in the Great War and then the Empire turned around and outlawed Talos worship. Ulfric finally did something about it, and he proved that Toryyg was not a true High King. The High King is suppose to be a strong and powerful man, not a young man who has little battle experience. Technically speaking, Ulfric is a worthy contender for High King, because he is true to the Old Ways and rightfully defeated Toryyg. He has done nothing wrong. Ulfric killed Toryyg to prove to everyone that he was not a rightful High King, merely a puppet instated by the Empire.