There aren't regional and national highways in Italy. All A-something roads are divided motorways with 2 or 3 lanes in each direction and no level crossings.
Most of A motorways are also toll roads.
In La Spezia you'd take the A15 to Parma to cross the mountains. It runs mostly in tunnels and on viaducts, that's why it's both easy to drive and expensive. Easy bearing in mind you're driving through the Apennines, not the Tuscan hills. After the A15 you'll take the A1 up to Bologna and then the A13 to Venice.
Use viamichelin.com for real driving times and overall costs. It shouldn't take more than 5 hours and half.
In Parma you could exit the motorway system and use local "SS" and "SP" roads to visit Mantua, Verona and Padua along the way. The Bridal Chamber in Mantua, painted by Mantegna, has recently been re-opened.