My guesses and my picks

Best Actor

Who will win - Mickey Rourke. Like I said below, Holywood loves a comeback.
Who should win - though it is unfair to talk without having seen Mickey Rourke's performance, I think Sean Penn's performance is Oscar material.

Best Supporting Actor

Who will win - Heath Ledger. And not because he died. In fact, Holywood has not been too keen on phostumous oscars (see Spencer Tracy and James Dean). But this time will be different..
Who should win - Heath Ledger. It is simply an amazingly complex character that Heath Ledger tackles beatifully and comically.

Best Actress

Who will win - Kate Winslet. After six nominations the Academy feels the pressure to award her and as Kate Winslet said in an Extras episode a couple years ago: Holocaust movies are always good for the Oscars
Who should win - Kate Winslet is an amazing actress and it is not a shameful choice that's for sure. But I only saw Anne Hathaway's performance (which is her best performance ever but it's not Kate Winslet level) so I don't know enough here

Best Supporting Actress

Who will win - This one is difficult. I'm going to guess Viola Davis (for Doubt)
Who should win - I have no idea. I only saw Taraji Henson (in Benjamin Button) and I don't think it deserves an Oscar. But Penelope Cruz or Marisa Tomei could also win the Oscar.

Best Animated Feature Film

Who will win - Wall-E
Who should win - Wall-E

Best Foreign Language Film

Who will win - Waltz with Bashir
Who should win - I don't know. I did not see any of it though I'd like to.

Best Director

Who will win - this will be very interesting to see. It's going to be tight. I'm going to guess Gus Van Sant.
Who should win - Oscars are a very complicated thing. They should be awarding the best performances in each year. However, the Academy's choices often mirror political issues and past performances. It was definitely because of past performances that Scorcese got the Oscar in the year of The Departed. So, this is a tough Oscar. On the one hand, I think Danny Boyle does a great work with Slumdog. On the other hand Gus van Sant and David Fincher are (for me) two special directors who have deserved an Oscar before and never got it. David Fincher has made three films that are always on my favorites list (Seven, The Game and Fight Club). Gus van Sant did Good Will Hunting (and many other interesting things). Moreover, Gus van Sant does an amazing work in Milk. Considering past and present, I consider that a deserving winner would be Gus van Sant.

Best Picture

Who will win - Slumdog Millionaire. Holywood loves an underdog and inspiring stories.
Who should win - The most enjoyable movie of the 2008 nominees is Slumdog, but considering everything I would say Milk is a better film.

I had more to say about the Best Picture Oscar but the Oscars are going to start in 3 minutes and I want to see if I'm right or not.