Florence - Stay in City or Country?

I've done both now, which four nights would allow you to do, but I prefer in town. If you like the galleries, it's really time and cost efficient. I stayed in a family apartment. We only had wine and coffee because it was cheap enough to let us explore restaurants. It's a relaxed city, not tense.
Full disclosure ... I did not like the agritourismo place in Tuscany. The beautiful countryside dissolved a shoe, the German-Italian owner preferred the German guests, etc. I was welcomed wonderfully everywhere in Firenze.

I would stay in Florence and rent a car for a day to go into the countryside. If you are city people, there is not much to do, especially at night, in the countryside.

End of April sells out fast due to two major holidays (4/25 and 5/1) so you must act quickly.
I would stay in Florence, if you prefer the vibrancy of cities. Lots of fun in Florence in the evenings. If you want to visit the countryside for a day, you can do so with a car rental for a day. Some towns can be reached easily by bus too, so it's up to you.
I'll give you the best of both worlds by suggesting a great location. Book a room at the Best Western Grand Hotel Adriatico on via Maso Finiguerra. It's a nice hotel (never stayed there but I know it because it is near where the Business School of the University of Florence, where I studied, used to be). Probably no more that €150-200 a night. All rental car offices are within 100 yards, in case you decide to rent a car for a day. The bus station (if you take the bus to Siena) is 500 ft away, and so is the train station. At the corner across from the hotel (corner of via Palazzuolo and via dell'Albero) is the cheapest trattoria in the area (that's what fed me lunch for my 5 years in school). Most major sights are within 15 min walk. Across the street is another Best Western, the Palazzo Ognissanti. I think the Adriatico is nicer and more modern, but if it's booked, try the second.

If you know the day when to rent a car, book it through a consolidator, such as www.kemwel.com or www.autoeurope.com. They get the best prices rather than directly with the rental companies.
In the area I told you above, basically via Borgo Ognissanti and via Finiguerra, are all the rental offices downtown Florence. To return the car there you must enter the ZTL (limited traffic zone), but it's allowed. Just request instructions on how to return the car when you rent.

Florence is a tiny city easily walkable if lodged fairly centrally. Commuting in from the countryside would be an enormous PITA. It is worthwhile to get a place in the countryside for a few days and use a car to visit small towns, abbeys etc but that is a different trip. We have done a weekend in Siena from Florence, driving the Chianti Road and then enjoying a day in Siena (can also be done by bus) You can't commute easily to FLorence by car and by public transport would be a hassle. Florence is a great 'being there' city -- want to go to FLorence? Be in Florence.

Consider on your next trip renting a place in the Val- d'Orcia for a week or so to explore that region.

My suggestion would be to stay in Florence, as there will be lots more to do in the evenings and also lots more choices for dining or other activities.

I can comment on Roberto's second choice for a hotel in Florence, the P Oggnissanti. I stayed there for several days a year ago with a church group. The hotel is fine and I wouldn't hesitate to send someone there. It was very much like a mid range American business hotel as I remember it – when I woke up I would have to remind myself that I wasn't in the Midwest on business. I wouldn't go back because I like lodging that reminds me that I'm somewhere different from home, but as I noted above I wouldn't hesitate to send relatives there. It was clean, the rooms were adequately appointed, and the staff was nice.

Florence has not many good connection to its suburbs or nearby country places. Stay in the city.

There are no suburbs in Italy.
Suburbs are an Anglo-Saxon invention, especially in America.
No town or village in Italy, no matter how small, will accept to be defined as "sub-" to any other city, no matter how big and famous.