Service Dogs can help people with disabilities become more mobile. It may seem intimidating to travel on an airplane with your service dog, but it can help to realize people who work on the airline see passengers with disabilities traveling with their service dogs all the time.
Most handlers call the airline ahead of time and let the airline know they'll be traveling with a service dog (because they may put a note on your ticket), but whether or not you call them, you go through security just like everyone else. Security will inspect the dog's vest and such, just like they do with people. The procedure varies, but often the handler walks through the security booth, leaving the dog, then calls the dog through, and then the dog is directly inspected by security. Sometimes the TSA does a hand swab on the handler, too.
When you arrive at at the gate and it's time to board, you'll board with the dog when you seat is called. You will probably then go to your bulkhead seat, which gives you a little more room at your feet. Some handlers with a service dog prefer to buy "stretch seating" or "economy comfort" so they get the extra room a bulkhead seat would provide, but also get the under-seat baggage room (this option costs money, but some handlers find the extra cost worth the expense). Some handlers even choose to pay the extra money to fly first-class. The dog then lies at your feet during the trip. The goal is to make everything as easy as possible - many passengers and flight attendants often don't even realize there was a dog on board!
You will probably feel more secure bringing paperwork (such as doctor's notes, the dog's service dog graduation certificate, and immunization records), but the airline often doesn't ask for any of this paperwork. Optional: Many passengers find it helpful to have medical documentation as a way to discreetly communicate information about their needs to security, and so the TSA has created a Notification Card that passengers can use for discreet communication. The URL of the card keeps changing, but you can go to https://tsa.gov and search for it there. Use of this Notification Card, or of a doctor' letter, does not exempt a passenger from screening, but the card can be a nice reassuring way of smoothing out situations.
We've always recommended that you do not get a connecting flight because would add too many hours of stressful travel. Plus, you would need to bring the dog outside to pee (and find a spot for them to pee outside), and then go back through security. Instead, we've advised that a direct flight is the way to go. At one point, a Pawsitivity family told us that they prefer a connecting flight because when traveling with a child with a disability, they found that two 2-hour flights are more manageable that one 4-hour flight...however, they have since changed their mind and now agree with us that a direct flight is (by far) their best choice.