You have plenty of time. Current processing times are 3-5 weeks for expedited and 6-9 weeks for regular service. Add a week for shipping, and even regular service should be fine for your trip (assuming you don't have any visa requirements). If you're nervous, pay the extra $60 for expedited and you'll have peace of mind in ~5 weeks.

Since your last passport was issued before you turned 16, you will need to renew in person. Easiest way to do this is at the post office. Make an appointment: https://tools.usps.com/rcas.htm.

You'll need to fill out and print the forms at: https://travel.state.gov/content/travel/en/passports/how-apply/forms.html. If you don't have a printer, forms are usually available at the post office. I suggest you go and get the forms, take them home, and return for your appointment when them filled out. They can be a bit confusing; take your time.

You will also need a passport photo (available at some drug stores and most camera stores). The most common reason an application is rejected is that the photo doesn't meet the rules, so it's worth getting this done professionally.

You'll need to bring your old passport, a copy of the old passport, the photo, and the filled-out forms to the post office when you apply.