To cut or copy any text, you must first select it. In most apps (such as Chrome), you select text by long-pressing. That will then reveal two sliders to set the beginning and the end of the selection. While selecting, a menu bar will appear at the top of the screen with icons for the copy, cut and paste tools.
When you are in a text editing box (eg the address bar in Chrome, or the text entry field in Messaging), long-press on empty space and you will get the option to paste text. To copy text messages, long-press to copy the full text (partial copying is not possible).