Need some Tuscany/Wine Country Advice

Bologna is between Venice and Florence. I've not been there, but I've heard great things.

Would it be better to take a train to Florence or Siena and rent a car from there?

You will want to rent a car for this leg of your trip. The Tuscan countryside isn't conducive to train travel, especially if you are wanting to get out to wineries. I think that this is a great idea to mix up the pace of your trip. We always try to do this as well.

I would suggest taking a look at agriturismos. These are a very fun and usually inexpensive way to travel through the Italian countryside. Many times they produce their own wines.

As far as wine regions, you won't go wrong with any of them. Is there a particular type of wine that you are wanting to explore?

Ok, so we're thinking of staying in one of the agriturismos in Chianti. If we do that the best idea is probably to take the train from Venice to Florence and then drive from Florence to Chianti. Should we then back track to Florence and take the train to Rome or drive to Rome (returning the car at FCO probably)?

This is all very new to us!

With your itinerary, it's probably easier and cheaper to rent in Venice (at Piazzale Roma) and return near Rome. This way, you avoid paying a train fare the same day as a car rental. Also, since you have no desire to visit Florence, this avoids the major hassles associated with driving there. Just watch your speed on the causeway leaving Venice (speed cameras will lead to a ticket, months later, if you go even a bit over the limit, according to other posters here).

There is another thread with tips from Roberto on returning a car near Rome, while avoiding ZTL's. If you can't find it, and if he doesn't answer here, you can send him a private message.

renting a car would be necessary, otherwise you will not be able to reach any vienyards or cellar in the countryside

I might suggest staying somewhere near Siena and visiting as many hill towns as you can. Volterra, Montepulciano, and Montalcino were some of our favorites and Siena itself is a beautiful city. The drive from Montepulciano to Montalcino has fantastic views. Pisa (west of Florence) is fun for the Tower grounds, but don't need to plan a lot of time there. If you stay further north, one of our other favorites is Lucca (also west of Florence) and it's not mobbed with tourists like a lot of the hill towns. All of the agriturismos/villas that we've stayed in make their own wine and it has always been fantastic, especially sitting on a patio in the Tuscan hills looking over vineyards and olive groves - sublime!

I spent a lot of time in Tuscany when I was studying in Florence about 7 years ago. I didn't drink wine then but I always loved food. The corridor between Florence and Lucca is chock full of places to go to for food and wine.
I would recommend you take the train from Venice to Florence and only get the car when you can't take public transportation to where you want to go or when you want to explore off-the beaten track.
You will have to change in Florence to get to Siena (best way is by bus).
And for what you want to do, a good base for touring would be Siena. Pick up your car there. I say this because I always think it's a headache to have a car in a city and try to park it.

Now what wines are good? In case you want to get an idea of some wines. Stop in at a wine bar (or two!) in Siena.
Find out the wines you like and plot out a route.

Colbertbr, you are in luck! It happens that I have a book called "Slow Wine 2014" which I picked up after attending a Slow Wine gathering of Italian winemakers here in San Francisco a month ago. fyi, Slow Food is a movement that was started in Italy advocating for the preservation of artisanal and "slow" techniques for making food. It also carried over into winemaking. If you can't get this book but just want a few recommendations from it, here's a few wineries that offer/make a variety of wines (all in the Chianti region of Tuscany):

(1) Castello di Monsanto (Chianti region) www.castellodimonsanto.it - chianti classico
(2) San Giusto a Rentennano (Gaiole in Chianti/part of the Siena region) www.fattoriasangiusto.it
(3) Rocca di Castagnoli (Gaiole in Chianti/ part of the Siena region) www.roccadicastagnoli.com - chianti, sangiovese
(4) Querciabella (Greve in Chianti/part of the Firenze region) www.querciabella.om - they blend various types of grapes to make their wines and have both reds and whites; recommended Batar 2011 chardonnay pinto bianco
(5) Poggerino (Radda in Chianti / part of the Siena region) www.poggerino.com - chianti classico, sangiovese and blended wines

Gaiole, Radda and Greve look really close to each other on a map.
These are just a few listings of the wineries in the Tuscany part of the book and they are selected by Italian wine experts choosing the best Italian wines. so I think they know their stuff.
Buon viaggio!

On a separate topic, if you have not yet done so, then reserve tickets ASAP for your train departing Paris. Are you taking the overnight (www.Thello.com)? Or the daytime TGV with a connection in Milan? Rick’s France rail travel page has the link and tips for buying “Prems” rate TGV tickets through SNCF (cheap, reserved, nonrefundable, available three or four months ahead of your travel date) to print at home and pay with PayPal. It does not tend to work with US credit cards, unless you have an extra level of security set up. Tickets within Italy are easy to buy in stations or available in advance at www.trenitalia.com. Our US source also sells most of these tickets, but not the cheaper Prems rate in France.