"It's freedom OF religion, not freedom FROM religion!" You've heard that, right? Actually it's both.
The only constitutional reference to either is right at the beginning of the Bill of Rights. "Congress shall pass no law respecting the establishment of religion (freedom from) nor prohibiting the free exercise thereof (freedom of). So simple. You can have yours and I can have mine. And neither of us can impose his on the other, nor can the government impose any on either of us.
Regarding contraception, then. If your religion teaches you that using artificial contraception is 'wrong,' you are free to not use it. But you are NOT free to insist that I don't. Nor may the government do so, at least, not on religious grounds. See the first amendment. (Freedom from.)
If the issue is funding then let's work that out. If contraception is to be part of comprehensive health care plans (yes please) and you're worried that some of your money is going toward funding it, through premiums or taxes, then do what the rest of us do when tax money gets used to fund something with which we disagree. Something, even, that we see as morally wrong, like 'preventive' war. Tell yourself that YOUR money isn't going toward that. YOUR money is going to fund things that you whole-heartedly support.
And smile. ; )