I don't do 3D movies at all, so I can't speak from personal experience for any of this. That said...for 3D, something that was shot in 3D has a better chance of looking great in 3D over something that was shot in 2D and converted to 3D afterwards. Not that converted movies always look bad. Some recent releases that were converted actually look really great. Gravity is a great choice for a (mostly) native 3D movie, is probably the premiere 3D showcase movie, and is currently on sale for $20 at Amazon (but not for much longer). Pacific Rim was converted, but the conversion is supposed to be outstanding. I can't speak to whether Avatar is any good as a movie as I haven't seen it, but it was the movie that really kicked off the modern revival of 3D film and is still one of the better representations of the format. If you're a Dr. Who fan, The Day of the Doctor was shot in 3D and looks great.
In general, Blu-ray transfers always look better than their DVD counterparts. This is straight up fact for 99.99% of modern releases, and generally the case for old movies that were transferred over. I strongly suggest bookmarking
blu-ray.com if you have any interest in the format. They're great for keeping up with sales, and in general their reviews are representative of what you can expect from a given release. If you want to know how a movie's presentation fares in HD, you can find out there. As for recommendations from the personal collection...
Some older flicks that look fantastic on Blu-ray:
2001: A Space Odyssey
Blade Runner
Once Upon a Time in the West
Se7en
The Nightmare Before Christmas
Taxi Driver
The Wizard of Oz
The Outlaw Josey Wales
all of the Godfather movies
all of the Indiana Jones movies
The first three Alien movies (could just buy the Anthology set, but the individual titles can regularly be found on sale for $5, and no one wants to pay extra to own Resurrection)
And some newer, 2D, ones:
Nolan's Batman trilogy (especially The Dark Knight and The Dark Knight Rises. Batman Begins is just a copy/paste job of the HD DVD transfer and doesn't look as good as the other two.)
every season of Archer
Sin City
District 9
The Departed
Casino Royale
Skyfall
Gravity (still looks immaculate in 2D)
Pacific Rim (ditto)
Moon
Sunshine
Band of Brothers
Deadwood
Cornetto Trilogy (Shaun of the Dead/Hot Fuzz/The World's End)
Kill Bill 1 & 2
any of the Marvel movies
all of the LOTR/Hobbit movies