There are 15 Trophies that can be earned in this title.

Bronze

Miles 'Tails' Prower
Become friends with Tails.

Knuckles the Echidna
Become friends with Knuckles.

Amy Rose
Become friends with Amy.

E-102 'y'
Become friends with E-102.

Big the Cat
Become friends with Big.

Sonic the Hedgehog
Clear the Sonic story.

Super Sonic
Clear all characters stories.

Metal Sonic
Get all 130 Emblems by playing Sonic Adventure DX.

Silver

The Fastest & Strongest
Get Level A on all the Action Stages.

Sub Game Master
Get all Emblems of the Sub Games.

Chao's Best Friend
Get all Emblems of the Chao Race.

The Adventurer
Get all Emblems in the Adventure Field.

Metal Sonic Master
Get Level A on all the Action Stages by using Metal Sonic.

Gold

The Perfect Adventurer
Get 130 Emblems.

Mission All Accomplished
Clear all 60 missions.

http://www.ps3trophies.org/game/sonic-adventure/trophies/