Actually, I usually go running in a ratty old t-shirt and satiny soccer shorts, circa my college years. But if I were more put-together, this is what I’d wear.
(These are the shoes I use, though, and the Amphipod for my smart phone, so I can track how far I’ve gone. And I do use a similar dog leash that has a belt – only mine has two leashes, natch! And I don’t buy special bags for dog poo – yet).