You can’t really go wrong with an MTB in that area. Finding trails is easy – there are forests all over the place, most with fire trails throughout them and loads of single track trails. No hassle from anyone for riding them either. Take a couple of different tyres, some semi-slicks and some for more rocky stuff.
I’d take the road bike though. You can ride the strada bianchi which are brilliant white gravel roads mostly the Chianti region, but stay on the tarmac and marvel at how smooth and well designed a back country road can be. The drivers are great giving riders more space than you will ever need. Italian clubbies are right friendly too. If you get passed by them they will call you in the bunch and try to teach you Italian.