Will be staying at the castle in April. What is the best way to travel from there to Florence, and other places. We do not want to rent a car, and would like some information on trains and taxi service. Thank you
Do you mean the Castello di Montegufoni in Montespertoli?
Luckily, Montespertoli is well connected to Florence by bus so you can take the bus #37 that has a stop about just 1km from the castle that permits you to walk the rest of the way. Take a look at the map here: https://goo.gl/maps/JLEtqkjsAfs
I would contact the castle directly and ask about local taxi service to use since they likely offer that service to guests at better rates than if you go looking for yourself - or can put you in touch with local drivers that offer the service.
Otherwise, with no car, you will have to take bus #37 back to Florence and then use Florence as your departing point for your day trips - you will be able to go wherever you want at that point, including doing group tours out of Florence to other parts of Chianti and Tuscany.
Do consider it takes about an hour to get back and forth between Florence and the castle so you will to add on that time to any trip you plan to do.
The guided tours out of Florence will be the easiest way to get to see many parts that require a car, including Chianti, Val d'Orcia, wine tasting tours in Chianti.
The cities - such as Pisa, Lucca, Siena - you can easily get to on your own with train or bus. I recommend you use Google Maps to map out the routes you're interested in, the system offers info on public transportation.