What size gyuto should I buy?

In a home kitchen I'd pick a 210mm gyuto. It's the sweet spot: long enough for a full slicing stroke, short enough to stay nimble on a normal cutting board and easy to control while you're still building your technique.

Step up to a 240mm only if you regularly break down large cuts — roasts, whole fish, big vegetable prep. The extra length means longer strokes and fewer passes, but it also means more weight and a bigger board; if the tip is hanging off your bench you're fighting the knife, not cooking with it. If you're mainly doing vegetables and portions, the 240 gives you nothing you'll use.

For a first gyuto, buy a well-made stainless one around 210mm in the low hundreds. Traditional carbon steels (white and blue paper steel) cut and sharpen beautifully, but they rust if you leave them wet — learn one variable at a time. Better steel is step two.