Usually, “to search for something” means “to seek something”. We search a place for a something: He searched the whole neighborhood for his dog. If we already know about the dog, we can say simply, “He searched the whole neighborhood.” Alternatively: He searched for his dog in the whole neighborhood. Sometimes, the word “for” is used with it’s literal meaning along with “search”, to introduce the equivalent of an indirect object: She searched the whole neighborhood for her neighbor, who was out of town. In the last example, she was attempting to find her neighbor’s dog, since her neighbor was away and couldn’t do so. Theoretically, we could say: She searched for her neighbor’s dog in the whole neighborhood for her neighbor, who was out of town. Also theoretically possible (but definitely *not* recommended): She searched the whole neighborhood for her neighbor’s dog for her neighbor, who was out of town.

When you say... "Search the internet" "Search the city" "Search the forest" ...you are giving a command to look around carefully. But you are not explaining WHAT you are looking for. When you say... "search for a good translation" "search for the criminal" "search for the treasure" ...you are giving a command to look for something specific. But you are not explaining WHERE to look. When you say... "Search the internet for a good translation" "Search the city for the criminal" "Search the forest for the treasure" ...you are giving all of the information. But please note, you don't NEED to give all the information in one sentence if the other people can understand what you mean. If a group of police are investigating a bank robbery and the chief of police tells them to "search the city", then they can infer that he or she means "search the city for the criminal".