Although, both sentences make sense and are correct, I would still go for the options below. ( This is how I would use) The context is very important. If you are going to look at a lot of information (be books or internet) I would write research. If it is just looking up at specific information ( say a person, few pictures, social media) I would write search. Hope this helps.

You may say," I will research information the internet." You may also say, "I will search for information on the internet." My guess is that you may want to use the former in an academic setting. Both sentences make sense.