Search something/somebody means to rummage, for example: The police searched the whole building. Search for something means to make a stufy, research, or to look for something: The police is searching for the missing persons

"Searching for something" is specifically looking for an item or a piece of information..etc. Ex: I'm searching the house for my car keys. "Search something" could be used when you are searching one item. Ex: I searched the house. In other words, these two phrases are pretty much the same, only the usage differs a little.

"search" introduces the PLACE where you look "search for" introduces WHAT you want to find

you search for something unfound. you dont know where it is, how it looks but you are searching for it. searching for and looking for is the preseng active tense. while you are doing the action you use -ing. searchING, lookING eatING cleanING.

I think you can also use "looking for" in the place of "searching for". ~ I'm searching about penguins and their habitats. "Searching about" is a trickier one, I don't actually see it being used often. I would use it when trying to find information about something. But it doesn't feel natural to me to use it in a sentence.