[verb て-form] + いく
ていく joins the て-form of a verb to いく, "go". Its most literal use is doing something and then going, or taking something with you as you go.
もっていく持っていく is "take" a thing and つれていく連れていく is "take" a person or an animal. いく is usually written in kana here, and in casual speech ていく shortens to てく: 持ってく.With a verb of change, ていく says it will carry on from now into the future. てきた looks back at a change that has led up to now.
With an ordinary action, ていく says you will keep at it from here on, often as a resolution.
With a verb of motion, ていく shows the movement heading off from the speaker.