Arrive early so you don't stress going through security. You need to remove your shoes so wear something that is easy to take off/put on. Any belts will have to come off or any heavy jewelry, car keys, etc. You can load your boarding pass on your phone so get the app for whatever airline you are flying. They will also text you updates about your flight so that's handy. Check your flight status before you leave for the airport. Carry on bag sizes are smaller now so double check that, but if you are going just for a day or two, you should be able to get everything in a small rolling bag and a backpack and not have to worry.
Check in online 24 hours before your flight so you can just go to the security line at the airport and won't have to go to the check-in line. Know the sizes for carry-ons or you will have to check, and pay for, your bags. Smaller is better!! Tag your bags just in case. Sometimes some jerks on flight will put their bags in compartments not by your seat and you may end up having to gate-check your bags. Sign up for the frequent flier club so you earn miles for your flights.
Bring food or snacks. The empty water bottle idea is great since water in the airports can cost $3 or $4 for a bottle.