Most of the following information (if not indicated otherwise) is taken from Duden - Band 7 - Das Herkunftswörterbuch, the entries weit and wider/wieder:
The three German words weit, wider and wieder go back to the Indoeuropean root * ̯ui-, meaning apart (auseinander).
The words wider and wieder come from a comparative * ̯ui-t[e]ro-, meaning more apart, farther away. The meaning developed to against, towards (gegenüber, gegen) and then to again, back again (zurück, abermals). Until the 17th century there wasn't any difference in orthography between wieder and wider. Then some scholars started to use wider for the preposition and wieder for the adverb. As we can learn in this Wiktionary entry, English with and Swedish vid go back to the same root.
The word weit goes back to the compound root * ̯ui-itós with the adjectival meaning diverged, gone apart (auseinandergegangen). The adjective weiter is (once again) a comparative of this compound; we can read on Duden.de that the adverb weiter is technically an adverbial comparative of weit ("eigentlich adverbialer Komparativ von weit"). As I understand it (and this is speculation), the meaning of the adverb shifted in phrases like "Gehen wir weiter." (=let's cover a greater distance) to "Gehen/Machen wir weiter." (=let's go on/continue). Thus, nowadays weiter does not only denote a greater distance but also the continuation of an action.