So when you get that initial offer, what should you say? First, always say you want time to think it over. Once you've conducted or verified your salary research, you can call back. Reiterate that you are excited about this position, and name some of the parts of the offer that sound good to you (the benefits, the start date, etc.).
If the initial offer falls below the range that seems fair for the position, you can then say something like, "Based on my research, $XX,000 is a bit low for [this type of position, this area of the country, my level of experience]. I'm sure we can figure out a fair compensation. It seems that $XX,000 [number slightly higher than what you think you can get] would be more appropriate."
If the initial offer already meets or exceeds your minimum, that doesn't mean you shouldn't negotiate. You just need to be prepared to mention some more detailed reasons you feel the higher number is more fair. Again, you're not being antagonistic or argumentative by doing this. You're trying to help lay out all possible considerations that need to be taken into account while the both of you hash out a fair number.
If they push back, offer to share the statistics you found. Some people will want to see them, while others will simply take this as an indication that you are serious and they shouldn't try to lowball you. When I've shared statistics before, I've put a few screenshots into a one-page document, saved it as a PDF, and sent it via e-mail.
It's still possible you may get pushback at this point — I've encountered a situation where the company has its own industry-specific salary database and will only go off those numbers. But at least at that point you're debating over two sets of statistics, not over some ambiguous idea of what you each think you deserve to make.
In this case, you can choose to walk away from the offer, provide reasons why you qualify for a higher tier of the salary information they're using, or accept the offer. You can also try to change to negotiating something else, such as vacation time or a telecommuting arrangement, that might be easier for the employer to grant.
You can save face when accepting an offer you've said was too low by framing it as a strategic decision: "Although this salary is lower than what I would expect to make, I am very excited about the unique experiences I will get working for your company [or some other benefit], so it's still worth it for me to accept this offer."