Hey Diane!
I stumbled upon this thread and I have to say that I am on the same journey as you right now. About 4 years ago I decided I wanted to watch AFI's (American Film Institutes) list of the top 100 films ever made. After I completed that I decided for my next challenge that I would try to find and watch all of the Oscar Best Picture nominated films (Although I am doing the 497 number trying to watch all 6 films nominated for a picture prize that first year) Anyways, as of today 1/3/2013, I have watched 407 of the 497 with 90 to go! However, I have already set aside East Lynne, The White Parade, & The Patriot as films I will most likely never see, although I still have hope that someday I may. So far the journey has been fun and something I look forward to "nearly" completing someday.
Like you, I am a huge cinephile and I have watched over 6000 films in my 27 years of life. My girlfriend and I get caught up in Oscar season each year and so I haven't missed an Oscar nominated film in years, but TCM is generally my best bet for finding the classics. I look on through TCM's online schedule a few months in advance and then jot down the films I want to DVR and watch in the future. On occasion I have rented a DVD from the local library, but if it weren't for TCM I don't think I would have seen half of the movies I have seen that were made pre-1970.
As far as my favorite Best Picture Nominees? I can't just name one, so...
1920's - Sunrise or The Crowd
1930's - I am a Fugitive From a Chain Gang or The Wizard of Oz (But '39 was amazing for everything!)
1940's - Rebecca or The Maltese Falcon or Casablanca (I Love Bogey)
1950's - Sunset Blvd or Bride on the River Kwai
1960's - Dr. Strangelove or The Sound of Music
1970's - The Godfather I & II or Star Wars or The Sting
1980's - Dead Poet's Society or The Right Stuff or Raging Bull
1990's - Shawshank Redemption or Good Will Hunting
2000's - No Country for Old Men or Inglorious Basterds
There are definitely a lot of films that I think should never have been nominated and hundreds of movies that I think were snubbed, but most I enjoyed for one reason or another although some were drawn out and boring.
Nominations I have disliked: Viva Villa! How the West Was Won, Cavalcade, Cimarron, Around the World in 80 Days, The Human Comedy, Anthony Adverse, In Old Arizona
Anyways, good luck with your continued journey. 31 Days of Oscar is coming up on TCM and is a great time of year to knock off a bunch of films from your list.
-Matt