When people say Sonic Adventure 3 they mean a return to that style of Sonic gameplay, as opposed to the "boost" style used starting in Unleashed, and the various gimmicks in Heroes and Shadow that, while still kinda close to Adventure 1 and 2, made them pretty different.
Personally, I would also want Adventure Fields back. When done right, like in Adventure or Unleashed, they can add a lot of fun, though hopefully with the advances in tech since Sonic Adventure we can have much bigger and more lively ones now. But even on the Dreamcast, it was a lot of fun to run around looking for Emblems, Powerups, and talking to all the NPCs, who actually had a really surprising amount of detail in each of their plots. All this said, I could understand if others didn't like them, since it's not really what you think of when you think of Sonic.
tl;dr: Go back to the old style of 3D Sonic gameplay. Other characters and Adventure Fields can maybe be included but aren't necessary.
'06 is the one that actually plays like Sonic Adventure, though. It's just not even close to finished or close to good. Heroes is the better game but it plays very different compared to Adventure, while '06 is the exact same style. It also improves on a lot of concepts from the other two games. Shadow still basically plays like Sonic, only now he has more offensive capabilities while Sonic has more agility. (Shadow is also slow as shit, but then that's one reason it's a shit game.) Tails actually gets to explore the level since he's not constantly in a race now. Knuckles still has exploration sections but also more classic "get to the end" sections. To this day I contend that if Sonic '06 was finished, it could have been the best one. Instead we got an Alpha Prototype that didn't even have time to hide the fact it wasn't finished.
Also, Unleashed's Japanese name is Sonic World Adventure.
THIS. The extra missions and ranks in Adventure 2 were the best part. Some later Sonic games tried to do stuff similar to this, but none got it quite as good. Adventure 2 was helped by having a lot of unlockables spaced apart just right. You unlock something every time you beat all missions with any character, then when you get all A Ranks with any character, then when you 100% the game. And the unlockables get better as you go on. First alt costumes, then new characters (sure they're really just alt costumes but it's still cooler), then an awesome secret level. The unlockables in this game are really underrated.
This too. Fuck boost. Adventure did a fine job translating the 2D games to 3D. Boost sonic is a totally different thing. it's more just a translation of the shitty Dimps games.