Guidestar.com and charitynavigator.com are both good places to start. They provide expense ratios based on publicly filed statements.
DH and I concentrate most of our giving on our church and the local women's shelter. In both cases, we know the people in charge of spending the funds. If there's an organization where you volunteer (or you know people who volunteer), you're closer to the operation and can have more confidence that your money is being spent wisely.
We haven't had a lot of problems with solicitations from charities selling our information to other organizations although some of the places we've donated to send a LOT of repeat mails/e-mails. I do charity bike rides and just donate rather than hit my friends up for donations, and that gets me a lot of repeat solicitations.
There's a separate thread on donor-advised funds and if you have the means to create one (mine is with Fidelity), it's a great way to donate anonymously and still have the proper documentation to take a tax deduction. In my case, I can tell Fidelity to send them a check and say it's from me, or I can tell them to send the organization a check and tell them the donor wishes to remain anonymous. I love being able to do that.