We always book through the airline. You should keep an eye on fares and when they drop, go ahead and book. This should be several months before your trip.
Other tips for a first time flyer...
The worst thing about flying is airport security. Remember, you can have two carry-on items, one smaller item (purse, backpack, laptop case, etc) that can go underneath the seat in front of you. Non-medical liquids, gels, and aerosols, are not permitted except in 3 oz or less containers that fit into 1 clear plastic zip top bag. Medical liquids should be properly labeled and need to be declared to a TSA officer. There are also other prohibited items. If you aren't sure, TSA has a list. They also have an app for smartphones and tablets with a "Can I bring..." feature. You type in the item, and it will tell you "Checked bag/Carryon" "Checked bag only" "Special Instructions" etc.
A lot of people like slip-on shoes for travelling because it's easier to get through security. I typically wear sneakers because they weigh down your suitcase if you pack them and they're easier to move around in. Whatever floats your boat; it's a matter of personal preference
During the security check, anyone between the ages of 13 and 64 will need to remove their shoes. Everyone will need to remove laptops and other large electronic devices and that quart size bag of liquids. Cell phones and tablets (anything small) can stay in your bag. Also, take off jewelry, belts, anything metal, and remove everything from your pockets.
When packing, I recommend putting electronics, jewelry, medications, and anything else valuable in your carry on. I also recommend packing a set of clean clothes and pajamas in your carry on in case something does happen to your suitcase. The times we've had this happen have been few and far between, but it does happen.
If you get a cold and you end up stuffy, take a decongestant before you fly. Your ears will thank you. Also, bring gum to chew on the plane. It helps keep your ears clear.
Make sure you drink plenty of liquids before your trip. SW offers complimentary soft drinks. They'll come around once, but if you want a second one, just press the button for a flight attendant. Buy some drinks at the airport to take just in case there's turbulence and flight attendants cannot leave their seats.