What do I need to do to organize a peaceful political protest at my state capital?

Read first about the best of peaceful protests in history - the Satyagraha.
Its power has shaken the roots of Bristish empire. Mahatma Gandhi brought
Independence to India.

http://en.wikipedia.org/wiki/Satyagraha

http://www.mkgandhi.org/nonviolence/birthsatyagraha.htm

http://dfong.com/nonviol/basicsat.html

http://ssinha.com/satyagraha.htm

http://www.wildewildeweb.com/gandhi/

http://www.kamat.com/mmgandhi/satyagraha.htm

http://www.search.com/search?q=Satyagraha

Depending on the size, you may need to get a permit,

Yep, out in these parts you have o be approved for a protest of any kind with a permit, check with your city hall or Government Agency that handles permits, or looking into your City rules concerning this at the library.

Those are the limitations of your society. During the times of British rule in India,
Satyagrahis were put to all kinds of tortures in jail.

Now what we have is a free India. Any one can launch a
Peaceful protest anywhere in India. No permits are
Required. The police keep quite and sympathize with the
Protesters, and allow them even for weeks until their
Demands are met. Only when the protest turns violent,
Then some action is taken to prevent damage.

In which other country do you find such a government provision?