First, start with a Nesika Bay action ($1100). Then order a Hart, Shilen, or Lilja barrel (say $450 after machine work is done). Find a stock; HS Precision is okay, but not necessarily the last word ($600?). Jewell trigger (~$100). NightForce scope ($1100). MSP or Badger rings ($190). Assembly ($500). $3500 - that sounds about right for a precision rifle.
If you don't have a lathe, set aside $15000 (maybe more, maybe less) for one with enough stiffness and precision.
I'm trying to rain on your parade, but rack grade Winchesters, Remingtons, and Savages are not precision rifles. Some of them will be sub moa, most won't. The only way to make them precision is to dump a bunch of money in them, including part of the items (custom barrel, trigger, machine work, and qualified assembly) from the list above.
On the other hand, you can spend $500 or so for tools and rebuild rifles by buying prethreaded and chambered barrels, upgrading triggers, and swapping the stocks, and they will be pretty good rifles. But they will never equal a truly custom precision rifle with a truly great barrel, chamber, and crown.