twg,
It's difficult to provide a precise cost and time on the route from Siena to Monterosso, since these details will depend on which departure times and trains you use for each segment of the trip. To provide a "ballpark figure" on that trip, without the stop in Pisa, the travel time ranges from ~3-4 hours and the cost from about €16.40 - €23.60 PP.
Yes, there is a Deposito Bagagli at Pisa Centrale. However, keep in mind there can sometimes be a queue either for dropping off luggage or retrieving it, so if your train departing Pisa is an InterCity or Freccia and you miss the train that you've booked, your ticket is worthless and you'll have to buy another at full price (BASE fare).
I suppose you could walk to the Tower, but using a Taxi or Bus would be more time-efficient. The Pisa San Rossore station is closer to the Tower but alas, no luggage storage there.
Regarding your Aunt's suggestion, Rail Passes are rarely a cost effective option in Italy as ticket prices are very inexpensive. One other point to consider is that Passes DO NOT include the seat reservations which are compulsory on the fast trains. Those are about €8-10 PP and would have to be paid separately out-of-pocket. DO NOT be caught without valid reservations on a train that requires them or you'll face hefty fines starting at about €50 PP, which will be collected on the spot!
One factor to consider with the driver/tour guide trip at €350 (~US$385, not including gratuity) is that making that trip along with stops in Pisa and Lucca is going to take pretty much the whole day.
However, despite all the complications there is a relatively easy method to accomplish what you want via less expensive pubic transit. Here's the details.....
- Depart Siena 09:18, arrive Pisa Centrale at 11:03 (time 1H:45M, one change at Empoli) - both trains are Regionale so you MUST validate your tickets prior to boarding the train on the day of travel, or risk hefty fines! Tickets are €10.70 PP and there are no discounts possible on those - I'd suggest buying those tickets at the station when you arrive in Siena, a few days prior to travel. A departure at ~09:00 will give you time to enjoy the breakfast at your Siena hotel, which you have probably paid for.
- Upon arrival in Pisa, store your luggage and take a Taxi to the Campo dei Miracoli and Tower. As I recall, if you want to climb the tower, reservations are required.
- At the conclusion of your visit to the Tower, take a Taxi back to Pisa Centrale and retrieve your luggage, but leave lots of time for that.
- Depart Pisa Centrale at 13:42, arrive Monterosso at 14:52 (time 1H:10M, direct) - this is an InterCity so compulsory reservations - you can buy advance tickets on that (€9.00 PP if you manage to get "Economy" tickets), and I'd suggest using www.captaintrain.com as their website is very user friendly. However do NOT miss that train or you'll have to obtain replacement tickets (which will add to the time and annoyance factors).
If you'd like to spend longer at the Tower, the next departure from Pisa is at 14:48, arriving Monterosso at 16:16 (time 1H:28M, one change at La Spezia Centrale, Frecciabianca for first segment so compulsory reservations and Regionale for last segment so you must validate your ticket prior to boarding). Cost is presently listed at €11.70 PP but this will depend on whether you can get "Economy" tickets for the first segment.
All of this may sound rather complicated but in actuality it's relatively simple.