I'm another gun loving liberal* and some of this advice I consider more sound than others.
The good stuff:
For starters focus on firearms sports so they are aware that guns are NOT all for killing.
TAKE THEM SHOOTING.
Emphasize the skill involved in shooting sports.
you'll have better luck making them comfortable than you will trying to awe them
You already know to take them shooting, but I think the introduction to a shooting sport (like Sporting Clays, Trap or .22 Steel Challenge) is even more effective.
Stuff I'm skeptical about or disagree with:
Pass them a copy of GunFacts v5.1 and Dare them to read it from cover to cover.
You can't. Liberal thinking is based on emotional reasoning, not logic.
RE: GunFacts
I'm personally incredibly put-off by information presented as "FACT vs Fiction". I don't want a list of assertions, I want careful explanations and well reasoned responses to my concerns. This requires a respectful conversation moreso than being handed pre-packaged arguments.
RE: Illogical liberals
All liberals I converse with think carefully about their views and are very reflective. Many of them site John Rawls as very influential in forming their views. I most frequently see claims that they're categorically illogical made by people who cannot distinguish between logic and and their subjective "common sense".
I personally don't think these types of opinions are changed by well reasoned argument. Familiarity with shooting and firearms (and current laws) will pay off much more quickly than debate. Make it especially clear who prohibited possessors are.
It would be helpful to know the source of their discomfort.
If you want data, a great source is the book
Armed and Considered Dangerous: A survey of Felons and Their Firearms. It's old (pre-background checks), but answers questions like:
What demographic do violent criminals come from?
Are their crimes pre-meditated or crimes of opportunity?
What do they look for in a weapon (cheap, concealable, accurate, untraceable...)?
Why do they chose these weapons, or to carry weapons at all?
If they don't carry a weapon, why?
Where did they get their weapon?
etc.
*by Arizona standards definitely. I'm not sure I'd qualify in some other areas of the country.