For anything that has to be mailed, look into envelopes (and printing costs) first and size accordingly. The first time I designed invitations I didn't do that, and I ended up having to send them in envelopes that were way too big because I didn't want to spend a fortune on specially sized envelopes. Oops!
For birth announcements I usually do 5x7. If you're printing someplace that doesn't include envelopes, you can sometimes get cheaper envelopes in AC Moore or Target (the envelopes meant for card makers, near the scrapbooking or stamping supplies) than in a place like Staples. For other cards and invitations I do 4x5.5 so I can print at home and fit them into invitation envelopes (4 3/8 x 5 3/4).
As for weight, I print at home, so I have more options. The paper I use is Epson's Premium Presentation Paper Matte, and it's 44lb/9mil. It's a little heavier than regular photo paper, but is about the same weight as what my first son's announcements came on. (I ordered his, since it was before I digi-scrapped.)