I'm going to post directions to the State capital as well as places you can park.
Here is a map of the downtown area
Driving in from the West take I-40 and continue onto Wade Ave. If you stay on 40 east it turns right into Wade ave.
Get off Wade Ave at Glenwood Ave and head south. Turn Left onto Peace St. Turn Right onto Salisbury St. and that will take you past the Capital.
The Parking that is available is a huge lot on the NE corner of the capital. It's on the corner of Edenton and Wilmington.
There is another private parking garage that is free on the weekends in the First Union Bldg. This is the tallest bldg in downtown Raleigh.
There is also another parking garage on the corner of McDowell and Hargett next to the police station. This is only a couple blocks west and one block south of the Capital.
I hope this helps, if other FReepers have better ways of getting here for those coming from out of town feel free to post.
MKM