Laguage + Train question

To answer your train question, yes, you just buy one ticket (in advance at http://www.trenitalia.com/ or at the station on the day) and it will include both trains, On the ticket it will say the date, time, train number, class, coach number and seat for both trains.

There are many direct trains a day between Milan and Florence, making a stop in Bologna but not needing any changes.

The easiest way to get train route information for almost all European trains is the Bahn (German Rail) website http://reiseauskunft.bahn.de/bin/query.exe/en, following Rick's tutorial: http://www.ricksteves.com/travel-tips/transportation/trains/online-schedules. If you put in Milano Centrale and Firenze Santa Maria Novella as your stations, you will see many trains with 0 changes.

But just for completeness, let's say you want to go from Milan to Siena, which does not have direct train service, but will always require changes. You buy this as one ticket (no need to buy a separate ticket at the intermediate stations for the rest of the journey). Furthermore, you don't need to buy this ticket in Milan, but can buy it at any Italian train station or ticket machine. If you're using a machine, the default "FROM" setting is the station you're currently in - just be sure to change it. Similarly, if you're buying from a ticket window, it's good to have your "FROM" and "TO" written down, to avoid misunderstandings.

TRAINS:
Where are you looking online? You don't need to change trains in Bologna.
Florence is along the Milan-Rome line. Some trains from Milan to Rome make no stops along the way, but most trains stop in both Bologna and Florence along the way, however you don't need to get off and change trains. The website for the Italian train company is:
www.trenitalia.com
The stations you need to enter are:
MILANO CENTRALE
FIRENZE SANTA MARIA NOVELLA (sometimes abbreviated with SMN on timetables)
There is also another train operator that operates high speed trains on that line. It's called italotreno:
www.italotreno.it
This company uses the 'Milano Porta Garibaldi' station as their base in Milan. The Florence station is still SMN for both companies.

LANGUAGE:
You can use the informal "Tu" form (as in "Come stai?) with friends, relatives, and with young kids.
You should use the more formal "Lei" form (as in "Come sta?") in all other encounters. Therefore, unless you have friends or relatives there, you should use the formal "Lei", with practically everybody in your case (hotel staff, restaurant staff, taxi drivers, shop assistants, ticket vendors, etc.).
I noticed however that the younger generation tends to use the more informal "Tu" (especially among people of the same young age) more liberally than old Italians like me would have done.
However foreigners who try to speak Italian will get a pass when they use the informal "Tu" form even when Italians would consider it inappropriate. Italians understand it's a difficult language enough even without the intricacies of the formal and informal form distinction, which is typical of Romance languages (Spanish has similar rules with the use of "Usted").

One minor correction - as of December 13th this year, Italo Treno has moved from Milano Porta Garibaldi to Milano Centrale for all their trains to/from/through Milan. ItaloTreno biglietti

"Italo Informa che dal 13 Dicembre la stazione di Milano P.Garibaldi sarà sostituita da Milano Centrale."

As far as your language question, definitely rely on Roberto's response, as he is a native speaker. I use Lei with strangers. However, I've noticed that the request to move to tu comes very quickly from even casual acquaintances, which surprised me.

gr,

As the others have mentioned, there are numerous direct trains from Milan to Florence every day, and you can either buy a ticket when in Italy at either a Kiosk or staffed ticket office OR buy a ticket in advance online with either of the rail networks that offer high speed service. Tickets purchased in advance often come with substantial discounts. If you want to purchase advance tickets, you can use www.capitainetrain.com as they sell tickets for both rail networks in Italy and have a very user friendly website.

Note that tickets for the high speed trains have compulsory seat reservations (which are provided with the tickets), and these are specific to train, date and departure time so can only be used on the ONE train listed on the ticket. If you board the wrong train by mistake or whatever, you'll be deemed to travelling without a valid ticket and subject to hefty fines which will be collected on the spot! The same fines apply to those riding Regionale trains, Buses, Metro (which don't require reservations) and don't validate their tickets.

Be sure to note the station names as you'll need to know that when purchasing the tickets.

What I haven't seen noted is the assumption that you are already in Milan Center City. However if you are flying in to Milan there are two Airports, Malpensa (MXP) and Linate (LIN). Getting to Milan Central Station from LIN is relatively easy and cheap. You can take a public transportation (City Bus) or a Taxi (do not have current rates). However, Malpensa is about 50Km from the station, and the most efficient way to get there is the train. You can pre-buy your tickets at http://www.malpensaexpress.it/en/
which is separate from the trenitalia site above. I would like recommend pre-buying but I believe the tickets are for a specific time, so be sure to allow enough time to get off the plane, assuming no arrival delay, get your luggage, clear customs, and then haul your stuff down to the train platform. This is relatively unlikely, so plan on getting them at the train station near terminal 1. However since they run about every half hour, for a 30 to 40 Minute trip to Centrale, you can buy your tickets from a machine or the ticket kiosk near the platform.

One note, this is a commuter train, and not built for comfort. Last trip to Italy, I wanted to aquatint my wife and our friends traveling with us the coolness of European Train travel. Before the trip I expounded on the comfort and speed and on-timeness of intercity trains, as we usually fly between cities in Europe.

Well we just missed the train when we got to the platform, and by the time I bought the tickets and were just getting off the escalator the train waiting there, not a commuter, pulled out. There evidently was some trouble on the line so the next train didn't arrive for about 50 minutes, and then the 40 minute trip took about an hour.

The seats were these kind of benches no back rests, kinda like a low pommel horse. They were set running fore and aft in the car so you sat sideways, and hard as rocks. Every time the engineer put the brakes on, and not too gently either, we and our luggage would try to migrate to the front of the car.

No one said anything, but it wasn't till we got on the Milan - Venice InterCity and started moving at 100+ KPH in typical Intercity Comfortable Seating and had a light lunch on the train that I was even close to not getting the "evil eye" and was forgiven. BTW they were all blown away by the sheer size, scale and stone carvings of the Milan Central Station. Leave your self some time to explore this unique station a bit. Definitely, a monument to Italian Nationalism of the Mussolini era.

Thanks all for the advice!