To move your capital, double click your new capital and select the show trade summary button in the bottom left. Once in there, click the middle button of the three buttons in the bottom left corner.
To view senate missions, go into the Factions button (bottom right hand corner on campaign map) then clicj on the Senate Tab (upper right). Go into Senate Missions tab. It will show all current Senate missions, the reward, and the time left.
Moving your capital increases unrest due to distance from capital, as it is no longer your capital. But, it will increase public order in the new capital. It is best to have your capital in the center of the your empire to reduce distance to capital unrest.