Thank so much for replying, Toritoribe.
Still have 2 questions following your reply.
Since 勉強 is kind of vague in this case, then if I use suru-verbs such as 散歩, which probably won't be modified in most cases, are these two sentences identical too: 学校へ散歩しに行きます、 and 学校へ散歩に行きます
And, for sentence like サッカーに行きます, it's basically very vague, and can be interpreted in many ways right? Does it just mean: I went for soccer, which can mean I play, watch, practice, etc?