Have a look on the EE... There's a lot of nice 223 to start with.... Around 1000$ with correct scope on them.... I have some remmy and some savage and they are both good rifle to start with. So don't stick to a brand...go with the best platform for the price you Will find.... Check for a good stock too, not only for the action...pretty much any factory savage or remington Will be a 1MOA rifle.
Honestly, I think You should start with a .22lr to learn and go with a 308 after, but that's only my opinion..... 308 is a really nice caliber for reloaders..... I don't think there's anything wrong with a 223 either.
I started with a 700 5r milspec in 308...with a vortex crossfire scope...it was a 1MOA rifle
After I went into reloading....it was a .50MOA to .75MOA rifle and I gained a lot of consistency
Upgraded the setup with a Sightron SIII scope and EGW rings after a while...I was able to group .50MOA more often and did some group in the .25MOA.....
Then a Jewell trigger...gave me more consistency.......keeping it pretty much always under .50MOA...
So What did I learned? The best thing to improve accuraccy is reloading...after you must see where You are aiming clearly.....after that You gained consistency.....it's no more a "best group" but an average.... First thing you have to do if want maximum accuracy is to start reloading....
And after all that you start to go custom

. Because a tuned Ford will always have some weakness of a Ford.... P.S. You can change the word ford by any brand you want
Some good platform you can find on the EE with a decent scope for around 1000$:
Savage Benchrest
Savage FTR
Remington 700p
Remington 700 5r