EDIT: just noticed the OP has already got a 10/22. Nice choice, I enjoy mine! I will leave my post below, because why not. lol.
Everyone has their opinion. Here is mine.
Technology helps a lot of things, but sometimes it lets us create items that are technically superior, but in practice may not be as usable as their predecessor.
The 17 has a better ballistic coefficient. Does that make it a better cartridge?
IMO its improved BC is not enough to make up for its smaller mass. On a windy day at 100m, which should be well inside the 17hmr wheelhouse, I will still put my money on a 22lr to print a better group.
Then there is the cost. Beyond any discussion of caliber, trigger control will be the defining characteristic of how good a shooter you are. You will likely be able to shoot 10x the 22 for the money. Practice practice practice.
Regarding the firearm, I am a big believe in bolt action. The safety during the learning phase is undeniably better with a bolt action. A bolt gun will cycle any ammo. A bolt gun will get you out of the rapid fire mindset, which can be fun, but not a training tool.
This leads me to 22lr bolt gun. Buy the best you can afford, as it will be around forever. Walnut and blued steel.
C