Greetings!
I don't see anything wrong with the grammar in your first sentence. And certainly, "surf the internet" has become an accepted part of the vernacular. It does depend somewhat on what you are writing it for, but I think if you tried to think of other phrases that mean the same thing, it could end up sounding a bit tortured: "I made a decision to perform internet searches on the world wide web"? That sounds stuffy and pretentious.
Unless my eyes deceive me, a and d above are identical. Therefore, a, c, and d are all correct. Choice b leaves out a word.
Best of luck in your studies!
Thanks,
Sarah, EssayForum.com
Sarah, EssayForum.com