Fun question.

The takeover vector that leaps to mind is remote code execution vulnerabilities on websites connected to important/sensitive systems. This lets you bootstrap from ability to make HTTP GET requests, to (partial) control over any number of fun targets, like banks or Amazon's shipping.

The things that are one degree away from those (via e.g. an infected thumb drive) are even more exciting:

• Iranian nuclear centrifuges
• US nuclear centrifuges
• the electrical grid
• hopefully not actual US nuclear weapons, but this should be investigated...

Plausible first attempt: get into a defense contractor's computers and install Thompson's compiler backdoor. Now the AI can stick whatever code it wants on various weapons and blackmail anyone it wants or cause havoc in any number of other ways.