My assumption is that I would have to get the precinct data and then aggregate them up to the new lines? I assume that precincts can't cross state legislative lines, but don't know if that is accurate. If any of you guys have advice or know a tutorial I could use, that would be great!
Yeah, that's the gist of it. If you have shapefiles for the precinct boundaries with your results joined to them, just do a spatial join of the precincts (probably their centroids, which works the vast majority of the time) to a district boundary shapefile and sum up the votes. NC is a great state for this because all the old precinct lines and precinct vote totals are on the NCSBE website (at least for the recent past).
It's possible district boundaries may cross precinct lines; this is almost guaranteed to be the case if you're matching old precincts to new districts. In that case, to get a better estimate, I typically use dasymmetric mapping: basically, break apart the precincts to a lower level such as census blocks, and then aggregate up using block population as a weight. This method also gets around centroid mismatches if you have bizarrely convex precincts or districts, but it takes more work, and block population is a little out of date as this point (not that big of an issue, but it is one).