I just started running this summer even though I have NEVER been a runner (or even wanted to be one!) and I'm loving it! I've been doing the Couch to 5K Program (C25K) and highly recommend it. The program eases you in with running/walking intervals until you get to a non-stop 5k in 30 minutes. I am signed up to run/walk in the Portland Half Marathon in October and am planning on some other milestone races before that, as well as after to keep my motivation up.
I would say, yes, buy new shoes now--it could save you a lot of pain later. Most good shoes are right around $100, but you can find them cheaper too.
As for your schedule, that depends on you, but I get out first thing in the morning (usually before 5am). I'm up before everyone else, it's a great way to get the day started, and nothing "comes up" to prevent me from running later in the day. Plus, the temps are nice and cool. :) I also work full time with 2 kids and plenty of other activities, and this is the best way that I've found to make sure I get the time for myself. Also, my husband is awesome about letting me get out for longer runs on the weekends, but I still go in the morning.
Running in the rain.....I don't know yet! I'll let you know tomorrow morning! Actually, I think it'll probably be my regular outfit: shorts, a tank and a long sleeve shirt. When it gets to really raining (winter rain) I'll probably have to modify that, but it should work for now.
The best thing is just to get started, but don't overdo it, or you'll get discouraged. That's why I love the C25K plan. It increases your fitness and endurance at a pace that challenges you, but is totally doable.
Good luck and happy running!