I have 2 grandchildren, 10 and 17 months old.

Yeah, I know it's a little early...

But, I'm interested on what people think of the following choices to get the shooting at grandpa's range.

The choices I've seen so far:
Cricket
Chipmunk
Savage Rascal
Mossberg 801

Anybody got experience with them?

I would like to chime in here, if I may.
The Short Answer: It is NEVER to early to teach a child the responsibility of safety.
Look both ways before crossing a road.
Don't put finger in electric socket.
Stop, Drop and Roll.
Don't play with fire.
And the Five Firearm Safety Rules.

For a first time .22lr rifle to learn:
A S&W M&P 15-22

I have been on this path before and have considered the same choices that you have listed. Taking into consideration the Length of Pull, the overall weight of the rifle, simplicity of function, little finger friendly, obvious "Safety" lever, ability to learn on traditional iron sights and the ability to add a simple red dot for learning open sights.
A few examples of what I considered in the cost and value of the firearm.

The Cricket / Chipmunk I ruled out quickly, as the rifle did not grow with the child and had no way to mount an optic, could not be suppressed for young ears, and did nothing more than what a Red Ryder BB gun could do.

The Savage Rascal was a good contender, overall, would not go wrong with it. Did not like the internal box-magazine for starters.

The S&W M&P 15-22:
A Lightweight polymer rifle that is more forgiving and durable for kids to learn on. The control functions of the AR platform are more user friendly and intuitive in location. The Safety is clearly visible, and an AMBI-Control from Troy Industries will work on it. The flat top 1913 rail allows many different types of sights and optics. The stock may collapse and pull out with ease to adjust to different sizes, no tools require. The barrel is threaded and a suppressor may be added to quiet the sound for young ears. The detachable magazines have a simple thumb button, to release it from the receiver. The magazines are easy to reload. One may load one round at a time or a full 5 rnd, 10 rnd, or 30 rnd. Magazines have a clear cut out to visible see if they are loaded or not.
Rifle is easy to take apart, and learned to clean. Rifle can be transitioned easily from a prone, sitting, bench, standing or slung with a good sling.
S&W makes a high quality product.
They make them in a number of different colours, and are easily hydro-dipped or rattle can spray painted to distinguish between different kid's rifle.
I could go on... and on on the merits of a good .22LR.

However, in the end, the child must feel comfortable in the handling of the rifle. What feels good to a person over 5' tall will not feel right in the hands of one who is not yet 3' tall. The Cricket and the 10/22 had a bolt that was difficult for my 4 year old to operate to show clear. The M&P 15-22 has a very smooth and easy pull to show clear.

The M&P15-22 will continue to grow with the child, and last well into their adult life as a plausible rifle that is fun to shoot.

I would be happy to go into greater details of my experiences with passing on the skill and traditions of marksmanship with you, if you would like. It is good to remember to have a fun time doing this, and not be over-pushy with them.... and just let them have a good (SAFE) time.

Also, the Red Ryder BB gun, make sure you get the Youth Carbine model, and not the standard Red Ryder that is for teenagers-adults.

The only one of these I’ve had some experience with is the cricket. I think the only reason it is a contender is because it is noticeably cheaper than say a 10/22 so if the kid doesn’t like shooting you have wasted less. Other than that, it has a goofy bolt safety and won’t be much fun for very long. As gnome said; it doesn’t have much over a BB gun.

Personally I would go with a 10/22. They’ll grow into it,and it is versatile enough that it will be something they’ll use for the rest of their life. Now you can get them with adjustable stocks for a bit more cash. I do like Gnome’s suggestion of the 15-22 though for the same reasons.

Alternatively when my nephew was born I bought a AR lower and his dad and I started putting parts in a box. He’s finally getting into shooting this year at 9 (just didn’t have the interest before) and will probably get to shoot it with a CMMG conversion bolt sometime this year.

At the age they are buying a BB gun for the “now” present and lower stakes safety familiarization might be your best bet. But I may be bias there because my dad bought a red rider for me pretty much as soon as they found out my mom was pregnant and it was waiting for me when I turned four. I used that for a couple years for walking around and learning not to muzzle people and then I got a bolt action Marlin at 7 or 8. I shot the BB gun a hell of a lot more because I could shoot it on my own/with less supervision.

I also tend to be pretty sentimental about my guns. I’m in my mid 30’s and still have both the red rider and the marlin so longevity/future use means a lot to me.

I've got a Marlin 80DL pre-microgroove that belonged to my grandfather, but it is an adult size .22.

I have a M1-22.

I have a Ruger Charger with a brace.

I've got a CMMG conversion kit and I have a AR Pistol that might bridge the gap to a full size AR.

The biggest thing I was concerned about was the weight.

I have a Chipmunk that I taught both my children to shoot with. It has a scope and shoots well. I felt that giving a young child a semi-auto rifle wasn't a good first step (Just my opinion) so I went with a single shot with the pull back bolt. The next step was called a Marlin Lil Buckaroo which is about halfway between the chipmunk and a full size 22. I learned quickly that giving them something with immediate feedback kept them interested, I didn't have any steel when my kids were young so I used balloons. They would get very excited when the balloon popped. My children are 31 and 35 now and both good shooters, they both have young children now that will probably be learning with the same rifles that their parents learned on. Start em early!

Target choices for little kids:
For the BB and .22 balloons are great! It shows an immediate response, are cheap to use and can be fun for them to. i like to use the tin coffee cans turned onto their sides. A BB will hit the inside and make a nice sound they can hear. Then the BB can often be reused. A .22 will pierce the tin coffee can, and that sits well into their heads, the difference between a .22 and a BB.

As great as it is to learn the basic iron sights... having a cheap red dot on a .22 allows for them to build up the confidence of shooting without the struggle of getting the sights right. This keeps it fun with out it being overly complex.
...much like learning to swing a bat at a hollow plastic ball sitting on a stand before learning to hit the ball being thrown at you...

The advantage of a bolt or lever gun would be using .22 Shorts to reduce the sound or range while learning.
The advantage of a semi-auto is having quicker follow ups and learning how to reload a magazine. One can always just put one round into a magazine... or get a single shot mag limiter.

The most annoying thing for me and balloons is picking up the rubber before the crows get them. Filling up paper milk cartons with different colour sands and hanging them is neat to.

Have fun trying out different home made targets. just be mindful of richochete off of hard surfaces.

For example:
The Red Ryder Youth BB carbine has a Length of Pull of 13.75".
The Red Ryder ADULT BB rifle has a Length of pull of 15".

The S&W M&P 15-22 Length of pull is adjustable from 11" to 14.25".
The Savage Rascal Length of pull is 11.25".
The Cricket Length of pull is 11.5".
The Mossberg 801 Length of pull is 12.25".

The M&P 15-22 will collapse down shorter for a begining and easily extend out for the shooter. No tools needed to adjust between different kids on the range, no need to keep track of different butpads in a box someplace. the kid can adjust it them selves. Given that it goes up to nearly 15", it can be comfortable used by an adult.

Might cost a little more upfront, but It is well worth for them to have there own that will last a lifetime and keep.

My son is 7. This past Christmas I bought him a keystone chipmunk. It's wood and blue (what he likes) and is extremely accurate. It fits him perfectly and he loves shooting it.

Our cheap targets are costco 48 packs of water. They explode and super cheap. I think for 48 it's like $3. He is very proficient with his .22
I went with the cricket as he wanted a conventional look, not pink camo, or that other stuff. The length of pull is perfect and the single shot is as safe as you can get.
He has had a red Ryder for several years and learned with it, and still learning with his .22

What I did when I gave it to him was tie it to a good memory. He and I went hunting last year and got a nice 8 pointer. I made a gunrack out of the antlers and had his .22 in it. I think he likes his gunrack as much as his rifle lol. He was already saying how it's room for another rifle on the antlers lol.
Long story short I'd go with a rifle that fits the child, and is fun to shoot. So what if they out grow it, there not that expensive. Look on the bright side by buying one that fits, you just created a Heirloom that your grandkids will more than likely use.