Hey bro,
I've been shooting consistently for 25 years now - from .22's at 12 to 5.56mm (which, by the way, is a .22 cal) in the Army to whatever I feel like taking out now (with 7.62x54r and 8mm being my favorites).
Maytime has pretty much nailed your question: The fundamentals of distance/accuracy shooting are the most important facets to master, so concentrating on them by removing the recoil/noise/expectation is generally how we train our new shooters. I don't just pull out my .45 and tell them to throw several mags downrange, they begin with safety, then stance, then sight, then squeeze... all on a .22. No one sends any lead out of a barrel until they have the basic tenets of safety and stance down pat.
You can get good shooting habits from a .223, but it's more likely that it will take you longer and the propensity is there to develop poor habits.
For the most part, it's a simple matter of muscle memory, IE: The more you shoot using good habits the better you'll shoot. Period. You're better off putting 500 rounds of .22 downrange than you are 100 of .223, so cost of ammunition is important. You do yourself a disservice if you limit your shooting because of cost considerations just because you like the idea of a larger round.
I can't afford "hundreds of..." 5.56 every day, so more power to you if you can.
My suggestion: Master the basics as cheaply as you can, which is the .22LR or Hornet, then save up for a nice .223 to begin mixing in. But get those basics mastered. Muscle memory and good habits = rounds on target.
Best wishes!