There's no such thing as a starter for Greek yogurt, as Greek yogurt is simply regular yogurt with some or most of the whey removed. This is usually done (at home) by simply dumping yogurt into some sort of straining device -- a sieve lined with a few layers of cheesecloth or a coffee filter works fine. (Alternatively, or in addition, you can add thickeners, like pectin or gums, or even powdered milk, but I'm pretty sure that would be considered cheating.)
If there's a specific brand of yogurt that you like, there's your starter, as long as it has active cultures. Just stir in about 3 tablespoons of yogurt per quart of milk.
It's not necessary to have a yogurt machine, although it will probably keep things neater. But you can use anything that will maintain the proper temperature for incubation, 100° to 110°F (38° to 43°C), for at least five hours (the longer you incubate, the thicker and more tangy the yogurt will become). I've seen people make yogurt in Instant Pots, microwaves, well-insulated containers (like a Thermos), even slow cookers and proofing boxes. ETA: someone -- Alton Brown, maybe? makes it with a bowl and a heating pad.
There are many, many internet tutorials with more details.
