The "best young core" discussion is kind of vague and somewhat meaningless, I'm going to go with "most likely to win a championship after 2011." So three more years for the current veteran teams before the new blood takes over.
1st Tier: Already have all of the important pieces.
1. New Orleans Hornets. They could win the title sooner than 2011, but as of now I would rank them as the best team for the foreseeable future. Paul, West and Chandler makes a great core now and they still have some room to grow. I also expect Julian Wright to become an important part of the team in the next year or two and maybe the 2nd best player on the team.
2. Los Angeles Lakers. They'll have a late-prime Kobe and Gasol with a potentially dominant young center in Andrew Bynum for a few years. It looks like they won't be keeping Odom and the rest of the roster is pretty fungible so I don't know exactly what the rest of the team will look like, but they should have enough. They're championship hopes will largely rest on Bynum, if he busts (unlikely) or continues to have injury problems (more likely) they move down to the next tier.
3. Portland Trailblazers. I am not as enamored with this team as some, but as long as Greg Oden becomes the dominant center that he should be, they have enough. Like the Lakers with Bynum, if Oden ends up spending most of his time in rehab they'll top out as a lower-seed playoff team (I also have greater concerns about Oden's future health than I do Bynum). The rest of the team is solid but definitely missing a major piece without Oden. I think Roy is a really good player and leader but I don't think he'll ever be one of the top fifteen players in the league. Aldrige will be an above-average starter but not dominant or an all-star. The rest of the roster looks like a bunch of solid role-players to me, although they should be able to add another major piece next summer through free agency.
4. Philadelphia Sixers. They have a full lineup of young players that I like with Williams/Iguodala/Young/Speights/Dalembert. The addition of Elton Brand gives them the solid, star leader they need who will still be relevant in a few years when the rest of the team catches up. The key for them will be one of the youngsters stepping up his game and becoming a clear all-star, my money's on Young. In the meantime, they should be one of the better teams in the east but not quite a title contender until someone else steps up.
2nd Tier: Need to add a major piece.
5. Miami Heat. Chalmers, Wade and Beasley should make a great 1-2-3 for the future, but they still need a dominant big IMO. With Pat Riley in charge, the lure of South Beach and plenty of cap-space, I don't think they'll have much trouble finding that guy.
6. Utah Jazz. Currently, I don't think they have quite enough to win a title and I don't think they'll have a great chance to add a big piece anytime soon. If they manage to keep Boozer next summer, I think they'll be locking-in to a team that will plateau just a notch below the top, similar to the Stockton/Malone era. They need a solid post defender to replace Okur in the starting lineup and better shooting from their wings. Their best chance to add a major piece is the unprotected NYK's pick they own, as they won't be getting lotto-picks on their own, nor will they have the cap-space to bring in a major free agent.
7. Golden State Warriors. I'm a homer.

But seriously, if Anthony Randolph can become a star, and I certainly see that potential in him, the rest of the team looks very solid around him playing the fast-paced style that Mullin is institutionalizing in Oakland. Even if Randolph doesn't become that dominant player, I still think that Ellis/Wright/Biedrins will form a great core and there should be enough flexibility for Mullin to add that missing piece.
8. Orlando Magic. They have one of the best building block pieces in the league with Dwight Howard, unfortunately Otis Smith has already locked-in the rest of the team, leaving them without much flexibility and missing a major piece. Basically, I think they'll need to get lucky to win a championship, but I'm not willing to write off Dwight Howard in the league.
9. LeBron James. He's the wild-card. If he stays in Cleveland they don't have anything close to a 2nd star and need a new GM. As best I can tell, they're plan seems to be to have enough cap-space in 2010 to grab another max free agent in the hopes of keeping LeBron. If he leaves, whatever team he signs with automatically becomes at least a 2nd tier team.
10. Houston Rockets. Both McGrady and Yao will be free agents in 2010 and I have no idea what they or the team will do, or how healthy they will be. Assuming they stay and are decently healthy, Houston will still be a contender for a few years.
3rd Tier: Limbo (No Order)
San Antonio Spurs
Chicago Bulls
Oklahoma City *
Detroit Pistons
Boston Celtics
Washington Wizards
Toronto Raptors
Dallas Mavericks
Phoenix Suns
Denver Nuggets
New Jersey Nets
New York Knicks
Charlotte Bobcats
4th Tier: Vote of "No Confidence"
Minnesota Timberwolves
Memphis Grizzlies
Atlanta Hawks
Indiana Pacers
Milwaukee Bucks
Sacramento Kings
Los Angeles Clippers