Yup, umbrella is your best friend, make sure you get a decent sized one. When I first moved here, the idea of carrying an umbrella for anything, let alone disc golf, was foreign to me. I had one of those little collapsible umbrellas that fit nicely in my bag, seemed like a good idea, until i realized that when I was using it, the runoff from the umbrella dripped straight into my bag.
I almost never wear a jacket, especially if I'm worried about my score. I haven't worn any jacket that didn't adversely affect my movement. I'm a big fan of a fleece vest when it gets nipply out though. Main thing I recommend is to stay away from cotton clothing, it'll just get soaked, stick to your skin and make you cold and miserable. Also, base layers are important, a set of thermal underwear will go a long way towards keeping you happy.
Last year I got a pair of gaiters, and I'm sold on them. If you notice, whenever its wet conditions, not necessarily even rainy, people's pants start getting wet at the ankles almost immediately, and the wetness moves upward from there. Having the gaiters keeps you dry, especially if you have to go tromping through bushes and shrubs.