I recommend:

Il Salviatino

Via del Salviatino 21 - Florence (Firenze)

If you’re looking coastal, check out Livorno. Definitely not Pisa. If you’re looking for more of the “Tuscan” vibe like you’d see in the movies, then Siena is a good bet. I've heard great things about La Spezia, but was unable to visit. For Cinque Terre, there is a hiking tour you can find online that is stunning, if you’re able. Enter into the UNESCO grounds via train, then hike along the cliff-side trails from town to town, then take the ferry back to your starting point if you wish. Take the time to walk the streets of each town and soak it all in. Each town is unique and has something different they’re each known for.

To be honest, I think Pisa is completely a tourist trap and would not even bother unless you just have to see the Leaning Tower.  I was not impressed.  Lucca, on the other hand, is a wonderful little town, and I would definitely recommend seeing it!

The drive from Florence to the Italian Riviera isn't very long. I would skip Pisa which is highly overrated. We didn't care much for Lucca either. 

Drive through La Spezia to the town of Portovenere on the coast and spend the night there. You can then tour the Cinque Terre villages by boat rather than train or car. That will give you sea views of the five towns which is when they're at their best. 

Our favorite destination in the Italian Riviera is Portofino. It is a jet setter destination with mega yachts in the harbor and extremely scenic. If you want to know how the rich live, go there.

If it were my trip, I’d stay over in Siena. It’s magical.

Tuscany is a (pretty large) region of Italy. Florence is in Tuscany so if you have stayed in Florence you have already stayed overnight in Tuscany! :-)

The 'Italian Riviera'/Ligurian Riviera lies to the north-east of Florence. You don't say where you'll be based there, which makes it more difficult to make recommendations.

Pisa could,in theory, be en route but imo it would be better to visit it by train as a day or half-day trip from Florence. Pisa's ZTL (restricted traffic zone) is extensive, quite complex and closely monitored by CCTV cameras, with fines automatically taken from the credit card you used to hire the car:

http://www.comune.pisa.it/it/ufficio-scheda/22067/Restricted-Traffic-Zone.html

So, rather than visiting Pisa by car I'd suggest you visit Lucca. Its city walls and narrow streets, its many ancient churches and its superb Roman amphitheatre (converted into a city 'square' during the Medieval period and still full of cafes and shops today) make it an excellent place to explore. Central historical Lucca (like all/almost all Italian cities and large towns and many villages) is a ZTL but the parking outside the city walls has easy walking access to the centre.

You could certainly stay overnight in Lucca. It has more than enough interest to fill more than just a short visit but, like Pisa, it's not so far from Florence that it will make your onward trip any shorter. 

So, unless you are staying to the south/south-west of Genoa it might be better to make an overnight stop in La Spezia, which has several excellent museums and is an interesting, vibrant city to explore. From there you could use the (frequent) trains to visit some or all of the five Cinque Terre villages.  Please don't think of exploring the Cinque Terre by car: village access is only for residents and there are few,if any, places for visitors to park outside the villages.

If you're aiming for somewhere east of Genova  you might think of an overnight in the city itself (it too has a fascinating and vibrant historical centre) or in Rapallo, Santa Margherita Ligureor Portofino.