。
Yuko got off to a really quick start in the race.
Literal
Yuko [topic-は] very quickly race [object-を] start-[past-した].
すばやく ('quickly, swiftly, nimbly') is the adverbial form of the i-adjective すばやい — a slightly more lively word than 速く. スタートする is a katakana-loanword suru-verb, very common in sports and business register despite the existence of native 始める. Mixed-script sentences like this are thoroughly natural.