Here is the full list of all 19 Rush Rally 3 achievements.
Finish a race without hitting anything
Finish a race without going off the track
Win a trophy in the Junior Career Championship
Win a trophy in the Amateur Career Championship
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
Win a trophy in Single Rally mode without repairing your car
Travel over 1000 km
Hit over 1000 objects
Finish first in Airstrike mode without getting hit
Finish first in Roadworks mode without hitting anything
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
Finish first in Traffic mode without hitting another car