September 16 Monday – The Clemens family left Chambéry for Turin by the fast express train, which Sam noted “makes 4 miles an hour—the other trains make only 3 1/4 . By 11 we were out of sight of Chambery.” Three hours from Turin, the train barely won a race with a team of oxen, Sam wrote. It took eight more hours to arrive in Turin, at about 7 PM. They took rooms in the Hotel d’Europe, which Sam noted had “wonderful rooms”. They went to supper and drank Barolo wine.

September 17 Tuesday – The family spent the day in Turin, shopping and enjoying the sights.
September 18 Wednesday – The family left Turin at 9:15 AM and arrived at Milan at 1:30 PM.
Sam’s notebook is full of things they saw in Milan, and observations on a host of items and situations. Some favorites:
I think the arcade system is borrowed from Turin.
Saw a starchy suit of clothes marked $9—doorway full of dummies dressed—stepped in to order one like the $9—nothing inside! The old man hauled in the dummy, stripped him & I ordered the clothes sent to the hotel.

September 19 Thursday – The Clemens party spent the day looking around Milan.
September 20 Friday – Sam (and probably the ladies) went to see Da Vinci’s “Last Supper” in the refectory of the monastery of Santa Maria delle Grazie in Milan. “If there is anything worse than the original, it is the 15 or 20 copies in oil & water”. They also visited the “great picture gallery” (Brera).
“There are artists in Arkansas to-day who would not have had to paint signs for a living if they had had the luck to live in the time of the old masters”.

September 24 Tuesday – The Clemens party left Milan and traveled north to Bellagio on Lake Como. They stayed at the Grand Bretagne Hotel.
Sam’s notebook:
“Rainy, sour, cold, dreary. Removed a screen in our room & discovered a regular fire-place—for wood. Right away we had the first wood fire we had seen since we left our own house. This made the day cheery”.

Sam enjoyed some valipolicella wine; he observed blind musicians at Bellagio with a little Russian girl passing the plate.

September 25 Wednesday – The Clemens party left Bellagio at 10 AM. They met G.K. Mayer and wife who helped them take the lake boat down to Lecco, Italy, where they boarded the train. They suffered another ten-hour trip and arrived at Venice at 7:30 PM.. The family had looked forward to Venice as a “relaxing interlude in their long journey.” Livy’s itinerary called for a three-week stay.

From page 267-8 The Life of Mark Twain - The Middle Years 1871-1891:

October 17, 1878: In Venice, the Clemens family had increasingly been pressed by visitors who discovered Mark Twain was in the city. After three weeks of this, they left Venice for Florence, a ten hour train ride. They stayed at the Hotel de New York.

From page 269 The Life of Mark Twain - The Middle Years 1871-1891:

October 21 Monday – Livy wrote from Florence to her mother:
Florence is much more restful than Venice, because we have no social demands—and one ought to know no one when they are visiting picture galleries—The Chamberlains are a perfect delight, they never tax us in the least they are helpful to us and are bright beyond expression.
Mr. & Mrs. Augustus P. Chamberlaine were friends of Ralph Waldo Emerson. The Clemens met them in Venice.

October 28 Monday – The Clemens family left Florence for Rome. The trip took 8 hours and they arrived at 4:30 PM [MTNJ 2: 235]. The party stayed at the Hotel d’Allemania. Sam noted the cost of the rooms, three coffees, one beefsteak and three “table d’hotes” (communal table, full-course meal) totaling 48.25 francs, paid at 5 PM.

There does not seem to be any information of what route they took.

Bædeker Central Italy and Rome (1869) 

From page 269 The Life of Mark Twain - The Middle Years 1871-1891:

NOVEMBER 11 MONDAY – THE CLEMENS FAMILY LEFT ROME AT 10:50 AM, AND RETURNED TO FLORENCE, ITALY AT 6:50 PM, WHERE THEY SPENT THE NIGHT AT THE HOTEL DE NEW YORK.
THEY WERE HEADED NORTH TO SPEND THE WINTER IN MUNICH, A 600 MILE TRIP WITH 36 HOURS ON SLOW TRAINS, AND FOUR OVERNIGHT HOTEL STOPS TO MAKE THE JOURNEY MORE BEARABLE FOR LIVY.
SAM’S NOTEBOOK: