I'll just add to Chris' response that this is the reason why you need to allow some extra time when you transit an airport which was your original concern.
-When you are flying to Rome (that's right?) you have 2 scenarios:
1. If you have a direct flight to Rome from a US destination such as Miami, Atlanta or New York, you will do passport control/immigration at this airport where they will stamp your passport with your entry date to the Schengen Zone, then pick up your bags and go thru customs. In Europe, customs is just 2 doors, one red, one green. Green is nothing to declare and that is where you will go as you will likely not be bringing in anything you need to let them know about. Your time frame here doesn't matter as it is the end of your trip unless your Insight folks are supposed to meet you at a certain time.
2. If you land somewhere else in Europe such as CDG or Amsterdam to change planes from your flight from the US, you will go thru passport control/immigration where they will stamp your passport with your entry date to the Schengen Zone, probably go thru security again, and then board your plane with just a regular airline ID check at the gate to make sure your name matches your ticket. You will then do customs which is just walking thru the green door when you arrive in Rome.
-When you are leaving Venice you will likely just have one scenario which is that you will need to change planes somewhere at a European hub. In Venice you will show ID with your boarding pass just like you do in the US when you go thru security and when you board the plane. When you get to the hub, you will head for your International departure gate. You will go thru Passport Control where they will look at your boarding pass and stamp your passport with the date you are leaving the Schengen Zone. You may go thru security again at some point. You may go thru extra passport scrutiny possibly at the airline departure gate.
Now, I know you are a detailed person and would like to be ready for whatever is to come. With these scenarios they will just happen. You will be in line, someone will stamp or check your passport. You might be funneled thru another line where someone puts an extra sticker on the back. You may have to walk thru an extra layer of security within the airport or at the departure gate. You don't need to look for how to do any of these things, it will just be set up that you have to move thru the area but it all takes time and this is why everyone is saying a minimum of 2 hours when you transit. Some people are more comfortable with 3 hours especially if it is an airport where you will need to do Immigration formalities either in Europe or US.
The bottom line, is don't allow the person helping you with your airline reservations to schedule you too tightly especially if it is where you will go thru Immigration/Passport Control coming or going. If they want to give you an under 2 hour time frame ask them if they have ever traveled to Europe and had to change planes. If so what was their experience time-wise. I am sure they will hate to hear you have been getting advice from online forums, lol!!