Correct way to calculate an average If I have the cost and number of days for several trips to Italy and I want to calculate how much an average trip costs to Italy per day. So my data would look like this:
Trip  | Cost | Number of days
  1   | 140  |       3
  2   |  40  |       1

Would I do (total cost of all trips to Italy)/(total number of days in Italy), or would I take an average of all (cost of trip B)/(total number of days of trip B)?
 A: This depends on what you are asking, it is akin to wanting to know if you desire a weighted average (weighted by the # of days/trip) or a simple average. 
It is possible, and even likely, that the cost of a trip for a single day differs significantly from the daily cost of a multi-day trip: people who are in Italy for a single day or two are perhaps more likely to pack their day with multiple activities and expenses, whereas folks with several weeks may choose a small # of things to see each day. I believe this would argue against a weighted average, because we do not want to inflate the effect of longer trips since it is likely that their behavior is independently different from the behavior of shorter trips. I would report an "average daily cost" by taking the latter approach you outlined. 
As an example, if I have 5 people who took trips with the final details below:
1) 20 days, \$600 total, \$30/day
2) 2 days, \$200 total, \$100/day
3) 1 day, \$100 total, \$100/day
4) 3 days, \$240 total, \$80/day
5) 2 days, \$180 total, \$90/day
Approach A: The average amount spent for all days spent in Italy was: (600+200+100+240+180)/(28) = \$47.14/day
Approach B: The average daily amount spent per trip in Italy was: (30+100+100+80+90)/5 = \$80/day
Approach A is misleading, of all 5 trips people took to Italy, only one had a trip as cheap as \$47.14/day. Meanwhile, the vast majority of people who took trips to Italy should expect to spend at least \$80/day, which is closer to the average in Approach B. 
You should also see the inspection paradox
A: It sounds like you want the average per-day cost of a trip, which is likely more useful than the average cost of a trip, since a trip could be 1 day, 1 month, or 1 year long.
To find this, sum the total cost of all trips, and divide by the total number of days for all trips. That give you the mean per-day cost, averaged over all trips. This is the first approach you outline. The second approach where you divide the cost of one trip by the duration of that trip also give you a mean per-day cost, but only averaged over that one trip.
