My mother went downtown to shop yesterday.

Literal

Mother [topic-は] yesterday, downtown [to-へ] shopping [object-を] do-[purpose-に] went

繁華街 (はんかがい) = downtown/shopping district (繁 flourishing + 華 flower + 街 street). 買い物をしに行った = 'went to do shopping' — しに is the purpose form of する.