They don't recommend jumping and weaving until they dog is 18 months due to growth plates. However, there are A LOT of things they can learn without jumping and weaving.
Flatwork exercises, which often get ignored, are a big one. Flatwork includes things like recall to heel flat and on a turn, pushes, pulls, front and rear crosses - to introduce some basic handing. You can also work on teaching the dog where his back legs are - a simple ladder will work for this; just walk the dog through a ladder flat on the ground. The goal is not to step on the rungs but between them. Wobble/Buja boards are also good for a puppy to get them use to things moving under their feet. Finally, you can use a 1x8 plank to teach them to walk with their feet on a board; just lift it slightly off the ground. You can use this same plank to begin to teach 2-on-2-off contacts. Tunnels/Chutes can also be introduced.
There is A LOT to learn which surprised the heck out of me when I started. Just watching it on TV it didn't look as involved as it is but once you learn it it becomes second nature. Everything I listed above is safe for puppies. We even a 6 month old Dobe and 9 month old Sheltie in my club's Intro to Agility class (which I am training to teach). Both were fine. Where we touched on jumps, we just laid the bar on the ground for the Sheltie and put it at 8 inches for the Dobe (because of height - she could still step over it).