The best way to get started on Twitter is to jump right in. Sign up for an account, start following @firstgiving and others who are interested in your work, and then join the conversation. Be sure to include a link back to your FirstGiving nonprofit page when you're talking about fundraising and to your nonprofit's website when you're talking about your work. See what you can learn from other people's activities.
For everything you'd ever want to know about social media for nonprofit organizations, we also highly recommend following Beth Kanter @kanter.