You're right where I was last year, but my first hunting trip was aged 36!! I moved to Colorado from the UK. I also got the advice to start with squirrels and a year on I completely agree, it's a great way to start. As a stand alone hunt it's great, long season too compared to big game (in Colorado), as prep for other hunting it's good too.
I've found hunting is much more than just walking through the forest with a gun! Clothing, food, water, navigation and equipment, getting used to the terrain and environment etc, other than the weapon, are all more or less the same as big game hunting. Then there's the regs. Hunting in Colorado is the first part of a lawyers degree. It's not bad when you get your head around it but I was mighty confused for the longest time about where I can hunt what and when.
I have a 870 now but as someone who has stood in your shoes last year, I would say if you're looking to progress to hunting deer or anything with a scoped rifle, buy a .22 with a scope. Learning how to use a scoped weapon is another skill, and it's entirely different to shooting at a range. A shotgun requires less from you for accuracy but when you hunt deer you need to have you skills down. A shotgun wont get you there.
IMO you owe it to the animals you hunt to be well practiced, and a sufficient marksman for a humane kill. Some people don't care about that, but I do. I feel good about hunting and taking responsibility for the killing the meat I eat, but I take no pleasure in inflicting an unnecessarily gruesome death on a living thing.
While range practice is important, the adrenaline of hunting can't be simulated. Getting the cross hairs on your quarry makes your heart beat fast. No target will do that. Getting a shot off on a squirrel at 40yards is a challenge, I'm told a similar challenge to getting one on a deer at 200 yards.
Range time will get you familiar with your weapon, improve basic marksmanship and safety, and teach you ballistics (bullet drop). When you're in the forest though, you don't have a shooting bench so you have to get out and learn how to steady yourself in less than optimal conditions. I've started using a sling or shooting sticks and try to brace myself against a tree or practice sitting/squatting comfortably. I've found accuracy is all about creating a stable shooting platform. I bought a bipod and at the range I was deadly, out in the field the grass was too long and it was useless. lol.
I decided to get tags and a 30-06 rifle this year after being invited to go Deer Elk hunting in November by a couple of wildlife rangers buddies. I felt last years hunting squirrels was good practice but still feel I'm moving on too early but what an opportunity to hunt with such experienced people. I'll be scouting the hunting area over the next few weekends and will have my .22 with me, so a dual purpose trip each time!
I was given a Marlin .22 model 60 by my in-laws and punched paper for a couple of years before realizing I was allowed to actually hunt, by myself, for real. So I went. I headed out numerous times for squirrel and rabbits but didn't see a thing. Slowly I learned the skills I needed and how to find my quarry and get close enough for a shot. My final trip out last year I got my first squirrel.
I spent the summer hiking and climbing as I always do but with an eye towards spotting the signs of deer, elk, and small game; identifying trees and shrubs etc. It's added a dimension to my other activities. My first trip out this year I got my second squirrel. I also saw a bunch of squirrels and should have got a couple more. I expect to be more successful this year. I cooked him in the mountains in a stew, with a potato, onion and seasoning. Eating my free range organic meat in the mountains sheltering from the rain under a huge boulder left me feeling very at peace with the world.
I'm of the opinion that going hunting (and fishing) is the prize, coming home with something is just a bonus.
Best of luck!