Go to www.travel.state.gov. There is a passport application "wizard". You can complete your application online, print it, and follow the instructions to mail it in. Fee for routine service is $110. You have the option of paying for faster service. The wizard will calculate your fees but you cannot renew online. Do this asap as busy season (those getting ready for spring break and summer) is about to begin. You will most likely not be able to travel on your current passport; most countries require at least 3-6 months validity after the end of your trip. On same website you can search for the passport requirements of the country you are traveling to.
If you're not leaving the country before Feb 2019, then just send in the mail your passport for regular processing which is 4 weeks. To check the fees visit travel.state.gov/passports for more info
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