Meeting, building relationship with new SO's kids

Reading the “Introducing your child/children to the person you are dating (after divorce)” thread got me thinking… A lot of what was said there applies to my situation as well, but from a slightly different perspective.

Background:
I recently started “seeing” someone new. It’s going quite well, and seems to have “potential”. (Gee, euphemism much?) He lives in another city, so mostly we see each other on weekends. He’s the stereotypical divorced dad who has his kids with him every other weekend. His daughter is 13 and his son 12. From what I hear, he has really good relationships with them both, and they also get along with each other very well.

He’s been telling them about me and they seem genuinely pleased that dad’s so happy. In fact, his daughter asked to speak to me on the phone the other day, and they want me to come visit on one of “their” weekends. Dad/SO is all for it at this stage, but I’ve been putting it off because 1) I’m super-cautious about seeming to be “horning in” on their time or relationship with their dad, and 2) it still seems a bit early to me, and I don’t want them to be in the situation of dad having a girlfriend that they like (presumptuous, I know), and then things not working out. I know this especially because I’ve had 4 step-parents (both mom & dad married 3x) over the years.

Looking ahead, there’s also that whole not-a-parent, not-just-another-adult thing which can be awkward all around, so I’d like to build a good foundation with them on fun, friendly and mutually-respectful terms. And, lastly, since we do the weekend thing, I’ll likely be staying at his house with them, so there’s what to say/do, if anything, about “sleeping arrangements”.

So, I’m planning on continuing my go-slow strategy for the foreseeable future, but it’s seeming more and more likely the day will shortly come when we’ll all meet.

Any suggestions or advice for the girlfriend?

I would think it would be a good idea to establish very early that you don’t hate their mom or see her as any kind of a threat or comptition. This can be done pretty simply, once you have met them a few times, just by asking after her: “How’s your mom doing?” kind of thing.

I would avoid expressions of affection too early: my mother-in-law-to-be told me she loved me the first night she metme, nad i have never been able to take herseriously since.