Here is the full list of all 19 Rush Rally 3 achievements.
Finish first in Roadworks mode without hitting anything
Win a trophy in Single Rally mode without repairing your car
Win a trophy in the Amateur Career Championship
Finish first in Airstrike mode without getting hit
Finish first in Traffic mode without hitting another car
Win a trophy in the Junior Career Championship
Finish a race without hitting anything
Finish a race without going off the track
Win a trophy in the Semi-Pro Career Championship
Win a trophy in the Rush Rally Career Championship
Complete the Rally Cross Game mode
Complete the Skill Games mode
Complete the Single Rally Game mode
Own an R-Class vehicle
Travel over 1000 km
Hit over 1000 objects
Finish first in Skill Gates mode without missing or hitting a gate
Finish first in Coin Collect mode and collect all the coins
Finish first in Turbo mode and collect all the turbos