Yes, perfect English and a common phrase. I need to clean my shoes. The children need to wash their hands before eating; I need to clean my teeth/brush my teeth. I need to go to the doctor; I need to take the car to the mechanic, etc.
In these sentences the word "to" belongs to the following word, e.g. "to" clean, to do, to make, to take. In the other European languages, the word "to" is a part of the verb structure, e.g. делать. думать, петь, чистить и.т.д. In German it is similar: singen/ machen/ kochen/ tun/ denken/ etc.
In English we use the word "to" and sometimes it is not essential to say the verb - I want to... I need to... I have to.... and we would understand what verb should follow, from the context. I need to... what? I need to do (the verb) we were just talking about. I need to do it/ make it/ telephone that person, etc. I need to clean my shoes. Do you need to? Yes, they are dirty.