Spoil them with gifts, especially if you don’t see them frequently. The parents have to say “No,” because, aside from budgetary considerations, they have to teach the lesson that you don’t always get what you want. Grandparents get to be the special font of toys that is visited occasionally, but never fails to deliver. Grandparents get to give two cookies instead of one, because it’s OK for that to happen at grandma’s house, as long as the child understands that this is ONLY at grandma’s house (and maybe grandma/pa’s cookies are homemade, and are especially desirable, and also more prone to going stale, so you have an excuse).
What you should never do is undermine any of the parents fundamental beliefs. If you raised your kids to go to synagogue on Rosh Hashanah and Yom Kippur, and to prepare for a Bat Mitzvah, but after that religion was essentially “done,” and the kids ate cheeseburgers and pepperoni pizza, but sometime in their twenties, they, or at any rate one of them, became very observant, and married another observant Jew, and their child has never had non-kosher food, it is not “cute” to sneak the child a McDonald’s bacon cheeseburger. It is not being an indulgent grandparent to say “We don’t cover our heads around here; if G-d wanted my head covered, he would have left me some hair. Haha.” It’s being a jerk. However, maybe buying the kid the cute basketball kippah to wear when he is over, that he wants, and that Mom said was to expensive, is a great idea. And maybe Mom says Hanukkah isn’t Christmas, but you can negotiate one gift for the first night (or some very serious gelt), and then a bunch of gifts for Purim.
If the parents are vegetarians, don’t sneak the kid meat. If they recycle, that doesn’t mean you have to do it at your house, but you don’t lecture the kid on why the parents have a stick up the butt for doing it.
I don’t personally have these problems, but these are all things that have caused friends of mine to curtail grandparent visits.
You don’t come across to me as someone who would do that, I’m just mentioning it because you asked.
The people who mentioned snuggling are wise.
If you can visit to help when the baby is new, that will be appreciated. If you expect to be treated like guests, then you have a short memory. Remember how hard a newborn is?
My mother couldn’t visit right away when my son was born, but she paid all our bills for a month; this allowed my husband to take almost three weeks off of work. You may not be able to do that, but I’m just saying that for the immediate present, the new parents will be exhausted. You will be craving reports and pictures, and they will probably not be forthcoming. Don’t pester. An occasional “Hang in there-- they sleep through the night sooner than you’d expect” text is great, though.
That’s about all I’ve got.
My mother was not a terrific mother, but she is an excellent grandmother. So, whatever relationship you had with your children will probably change, and most likely for the better now that you have parenting in common. Plus, you have a chance to be this “grandparent” thing, that is a different role for them, and someone who helps them in ways they didn’t expect, so they may appreciate you more. That may be a side effect you didn’t expect. You also may have a new respect for them as you finally see them as real adults.