You will first need to make sure that the first six chapters of the Arcade Mode are unlocked before proceeding.
Choose "ARCADE MODE."
Choose "Catch Me If You Can," and then "Event 3" (St. Petersburg, Pushkin's Places, 2 Laps) for the FOG requirement.
Choose "Drifto 101," and then "Event 1" (Quebec, Rampart Run, 2 Laps) for the RAIN requirement.
Choose "Rough Riders," and then "Event 3" (Tokyo, Tochomae Loop) for the SNOW requirement.
BAM. 25 gees. Enjoy!