If you simply take one flight from point a to point b, then you are off the hook and don't have to worry about extra issues that may affect you had you additional connections. Connecting flights, regardless of whether they are with the same or different airlines, entail extra rules and elements that you may want to consider. From making connections to will your luggage be checked through - connections are often a beast!
1. The Rule of Duty Free
Buying duty free alcohol or perfume? If you are connecting flights you must pack duty free liquids in your luggage if going through security check points again - usually after customs. So if you are traveling Frankfurt - Chicago - Sacramento, for example, you'll need to put it in your checked luggage after customs in Chicago.
2. The Gate Change Effect
Check your gate number when you arrive in your connecting airport. The airline rep may have given you a number when you checked in, and it may have changed in the interim.
3. The Bad Weather Syndrome
In adverse weather check to see if you will make your connection - if not, and it is say a once a day flight and you now need to overnight the airline is not required to cover your expenses for weather issues. This varies depending on where you are traveling in the world (Europe is more strict regarding weather events).
4. The Cancelled Connection Conundrum
In adverse weather check to see if your connection is operating, if not it could be a long wait at the connecting airport trying to get out.
5. The Pick-Up Line - for Luggage
When you are first checking in your luggage, ask if you need to pick up luggage - ie. clearing customs may affect time needed for your connecting flight.
6. The Terminal Switch and Run
Ask if your connection is out of the same terminal as the flight you arrive in on. I have seen lots of people miss flights even with legal connections (what the airline deems enough time - usually based on airport guidelines and airlines create minimum connection times) because of distance between terminals and gates.
7. Help is on the Way
If you need assistance to get to your gate check in as much of your luggage as possible. It not only makes it faster to help get you there whether you are on a golf cart or in a wheelchair, but safer too.
8. The Stroller Necessities
If you have checked a stroller at the gate make sure they have tagged it to be brought to the door so you have it for your connecting flight.
9. The Change of Venue...and Airport
If your connecting flight is at a different airport allot extra time and money. Inter-airport connections can occur between New York's JFK and LaGuardia, and London's Heathrow and Gatwick airports for a few examples. If you are connecting from one airport to another it can cost you not just time, but a cab or shuttle fare too.
10. Getting Through to Your Luggage
Make sure your luggage is checked through as far as possible.
If you are going from a legacy airline like United Airlines to a low cost one like Aer Lingus your luggage cannot be checked through, and neither can you so you need to give yourself a lot of extra time.
I have seen a passenger in her 90s who was booked on American Airlines to Boston connecting with Airtran and the flight to Boston was delayed - luckily the Airtran flight was too, but she was not aware of having to pick up her luggage and check it again, and to check herself in for the flight.
Make agents aware of additional airlines you may be connecting with. Air Canada and Delta are not partner airlines but you can check luggage from Atlanta - Toronto - Quebec City.