How much can you expect to spend?
Me, I'm Single, self emplopyed ... I could go out 6 days a week and shoot 30 to 50 rounds at a time ... that's 24x50=1200 rounds/month just in that one caliber.
My buddy is Married, with kids, an 8-5 plus commute M-F job ... He'd be doing well to get out every Saturday for a couple hours. Call it 3 outings a month @ 50 rounds each ... 150 rounds/month.
Very hard to judge how much time another person can dedicate to their hobby - and the more time you have to shoot, the more ammo you will use up. A rough Guess-timate would be 40 - 50 rounds per outing. Only you can figure out how many outings per month.
I'll also second the .22 statements ... Far cheaper to take it out for an afternoon and shoot a couple hundred rounds. If you are shooting < 1" groups at 100 yards with it, you're making progress and can relay those skills to your 308 and longer distances.
Don't knock the old .22 Cooey's either ... with the right ammo, they are a <$100 gun that will kill quarter sized targets all day long at 50 yards.
how many rounds of ammo spent on practice does it take on average to start getting better?
In theory? Just one

Every round you put downrange with purpose will make a small improvemnt.
In reality? No idea ... Everyone learns at their own pace, and some "inherent" bad habits can take longer for some to break than others. Expected Recoil Flinch is the toughest to overcome imho - even those that are used to shooting large calibers can have it and not even know it - the act of bracing for the recoil as you squeeze the trigger can be enough to ruin a shot.
Providing your equipment is up to the task, I'd think that@ 200 yards, dropping from a 6" group to a 4" group should happen in a few serious outings - especially if you have a "coach" to watch you and make constructive pointers. Add in another month of "maintaining" those good shooting habits and starting to make them habit.
The next 1 1/2" drop in group size may take a season, or may come in a month.
Getting consistent below 2" groups (1 MOA) will take the rest of your life to maintain.
The further you stretch the distance to target on your training, the more time will be required as well - get to the level you want at a realistic range (100 yds for .22 - 200 or a bit more in 308, then start walking the targets out and finding where your limits (and your gun/ammo limits) are. Then you are looking for answers to why.