This is often called the "super-comma" function of the semicolon: it acts as a higher "level" of comma to separate list items when at least one of the items contains a comma. The general rule is to use the semicolons as you would use commas with ordinary list items:
I arrange [X], co-ordinate [Y], and write [Z].
I arrange interviews through phone and film; co-ordinate the editorial team’s travel; and write blog posts, articles, and reviews of local London events.
(This should make it easier to see that you don't need the last "I" in your sentence, as John Lawler notes in his comment. You could also precede all three list items with "I", although that would be kind of fussy and hypercorrect. The important thing is to use the same construction with all three list items.)
The so-called Oxford comma complicates matters. Many people put a comma before the last item in a list, and many people do not; it ultimately boils down to taste and whether the style guide you're working with requires it. If you're one who doesn't use the Oxford comma, you might be tempted to leave out the final semicolon in a sentence like the one above, which is a bad idea. Sentences containing complex lists are complicated enough that you should always include the final semicolon to ensure that readers parse all of the list items properly without having to reread the sentence several times.