I helped a compatriot file her citizenship recently (DV winner that'd been living here for 5 yrs). Processing time depends on your field office. It's quite plausible for your processing time to take upto 6 months on average (NYC, Boston, Houston are seriously backlogged w/ processing time lasting upto 9 months).
You can download your form from here. http://www.uscis.gov/n-400. The entire process goes like this.
1. Filing.
2. Fingerprint (within a month). Try to pick up your "civics test" material during this.
3. You'll receive a notification saying you've been placed in line for an Interview.
4. Your status changes to "scheduled for interview".
5. You may receive a "yellow letter" after your fingerprint. It's a pretty generic notice that normally asks you to bring your passport and greencard to your interview.
6. You show up at your interview. During the interview, you'll be asked to verify that all the info that you included in the application is correct. Your immigration officer might go through your passport to ensure that you didn't spend over 6 months (in 1 trip) outside the US. You'll be randomly asked 10 questions (out of 100 that's included in the cd you can receive during your fingerprint). You need to get 6 of those correct. Most people get the first 6 correct at which point, IO stops asking.
As part of your English test, You'll be asked to read a sentence and write one, too.
Interview is pretty much a formality provided that you don't have any warrants and criminal history. If you do, then they will ask you to submit court dispositions. If you've got tickets/citations less than $500/ticket, you won't need to disclose it either.
7. Depending on your field office, you might be asked to take the Oath that very day. If you live in a big city, you'll be asked to wait for your Oath letter (normally sent out 2-4 weeks after Interview day).
8. On the day of the interview, you'll be asked to take a pledge of allegiance. Upon taking Oath, you'll be issued a certificate of Naturalization.
I went to her Oath ceremony last month and it seemed quite a moving experience. There were quite a few emotional attendees - especially from Africa, Bhutan, and war-inflicted regions.
After you've received your certificate, you can apply for your US passport.