Wanting a thing (ほしい), wanting to do something (〜たい), and going somewhere for a purpose.
N が ほしいです = “I want N” (a thing). The thing wanted takes が.
あたらしい くるまが ほしいです。
I want a new car.
ほしい is an い-adjective meaning “wanted/desirable”. The desired object is marked with が. It describes the speaker's own wish (or, in a question, the listener's).
Replace ます with たいです to say “want to (do)”: 行きたいです.
にほんへ いきたいです。
I want to go to Japan.
Take the ます-stem (行きます → 行き) and add たいです to express “want to do”. たい conjugates like an い-adjective (negative 行きたくないです). The object may take を or が.
Verb-stem + に + 行きます = “go (somewhere) in order to do …”.
デパートへ かいものに いきます。
I'm going to the department store to shop.
The purpose of a movement is the verb-stem (or an action noun like 買い物) + に before 行きます/来ます/帰ります.