Successfully complete one of the following tasks to get a trophy:

One for the Road (Bronze): Rank up for the first time.
Now ya Talkin' (Bronze): Advance to Rank 10.
NXP Challenge (Bronze): Achieve over 6,666 NXP in a single race.
Keep Diggin' (Bronze): Drive a total of 500 miles or more.
Tailgating (Bronze): Perform a draft of 500 yards or more.
Sprint to the Finish (Bronze): Win a race in Career Mode and get the full 195 points.
VIP (Bronze): Complete all three objectives for a single sponsor in Career Mode.
Car Stylist (Bronze): Create a custom Pain Scheme.
Last Man Standing (Bronze): Survive a 43 car eliminator race to the end.
Run the Gauntlet (Bronze): Win a Gauntlet Invitational Event.
The Champ is here! (Bronze): Win the Championship Showdown Invitational Event.
Finishing in First! (Bronze): Win a race crossing the line in first gear.
Give me 4 (Bronze): Perform your first 4-tire pit stop in any race.
Hotrod (Bronze): Create a custom tuning setup.
'Let's Go Racin Boys' (Bronze): Take part in your first online race.
Up on the Wheel (Bronze): Lead 5 consecutive laps in an online race.
Rubbin is Racin! (Bronze): Race online at every track.
Pedal to the Metal (Bronze): Achieve the fastest lap of the race during an online race.
Styling 'N' Profiling (Bronze): Win a race with a personalized Paint Scheme.
Groovy (Bronze): Master a track. Complete all the objectives on a Master the Track card.
Decorated Driver (Silver): Advance to Rank 20.
Saddle Up & Hang On (Silver): Qualify in pole position at every circuit in the game in Career Mode.
Victory Lane (Silver): Win a race at every circuit in the game.
Star of Tomorrow (Silver): Achieve a total of 100 fastest laps.
Race Rivalry (Silver): Achieve 20 rival victories. (Online Only)
Race to the Chase (Silver): Achieve enough points in Career Mode to move into The Chase, as one of the top 12 drivers.
Cup Contender (Silver): Achieve a top 10 finish in every Career Mode race.
Any Place, Any Time (Silver): Compete in every International Event.
MVP (Silver): Receive full sponsorship for your driver in Career Mode.
Win on Sunday, Sell on Monday (Silver): Win a race in all brands of car.
Pit Perfection (Silver): Perform a 4-tire pit stop, and go on to win the race. (Online Only)
Ride in Style (Silver): Collect 88 Paint Schemes by racing in Invitational Events and reaching rank 29.
Surfin' USA (Silver): Master every track in the game.
Name to Remember (Silver): Unlock every Gold Legends Coin in the game.
Kingly (Silver): Defeat Richard Petty's race win record of 200 wins.
NASCAR Legend (Gold): Advance to Rank 30.
Series Champion (Gold): Finish top of Career Mode standings and become the Sprint Cup series champion.
Boy Scout (Gold): Unlock every NASCAR Pin in the game.
Platinum (Platinum): Unlock every trophy in the game.

Additionally, there are two secret trophies:

Celebration Shot (Bronze): You photographed the driver celebrating on the car after a Career Mode race win.
Rookie Mistake (Bronze): You received a drive through or speeding penalty.