As a grandparent, chances are real good that you've been a parent and know about the generalities of different age groups and these can help you in your focus when spending time with the grandchildren. Here are some ideas based on those parameters:
Newborn - Toddler
This is a time of learning their own bodies, how they work and the effect they have on the people around them, so to help them along (beginning at the youngest and working up):
Sing-song games - Pat-a-cake, This Little Piggy, etc. (Repetition is comforting)
Body-part identification -"This is your nose - this is my nose."
Shapes and colors
Animal sounds identification -"Pig says 'oink'; Cow says 'moo'."
Small and quickly accomplished chores -"See how quickly you can pick up all your books and put them on the shelf" - followed by lots and lots of praise.
Kid - Pre-teen
This is a time to explore talents and to create. Pay particular attention to where your child's interests lie between ages 4-8. That is often where their talents also lie. This is also a time to really delve into reading:
Reading to the younger ones and with older ones
This is a great opportunity to get into chapter books. The most important thing about that is consistency. If you get involved in a great book with them, make sure to continue to the end. They will love being able to count on that. This is also something that can be done long-distance through the phone or Skype.
Art - Experiment with lots and lots of mediums with them
I have a grandson that became very, very good at creating things out of trash: cardboard boxes, lids, paper towel rolls, egg cartons, etc. Make finger paints. Sculpt with clay. Whittle a bar of soap. Let them get messy, and you get messy with them. Art is a great way to communicate.
Life skills - Make cookies with them
Or bread. Or sew on a button, make a quilt, replace a zipper. Teach them ways to streamline cleaning their room. Teach them anything that you know how to do. I remember soldering components onto a motherboard with my father and learning binary code.
Music - Introduce them to musical diversity and share your memories associated with particular songs
- Tell them the stories from your youth and all about your grandparents, etc. Pass down the history. Encourage them to keep a journal.
Teens - Young Adults
- When you are doing a puzzle together, you have the opportunity to speak to one another without eye contact. You might be surprised at how liberating that is and how much inhibition it removes from the conversation.
Old-school crafts from your day
-Tie-dying, batik, decoupage, etc.
Service, service, service
- Help them to love serving others by doing projects with them - yard work for an elderly neighbor, cookies to a shut-in, visit a nursing home.
- When they are ready; it is entirely possible that you have more patience than their parents, but clear it with them first.
- Another great opportunity to really talk and see local sights.
If you don't live near them and don't get to see them often, here's a trick my mother once did for a reunion. She did all the holidays in a week. One day was Christmas with a tree and small homemade gifts. She did Thanksgiving, Easter, a giant birthday party, 4th of July, Halloween, and Valentine's Day. So much cousin fun!