YOU SAID:

I walked into the shop to buy a cake.

INTO JAPANESE

私はケーキを買うために店に入った。

BACK INTO ENGLISH

I went into the store to buy a cake.

INTO JAPANESE

私はケーキを買うために店に行きました。

BACK INTO ENGLISH

I went to the store to buy a cake.

INTO JAPANESE

私はケーキを買いに店に行きました。

BACK INTO ENGLISH

I went to the store to buy a cake.