"BEST" is relative. What is more important to you, aesthetics, or performance. If aesthetics are important a Winchester Featherweight, Ruger Hawkeye or Remington 700 is where I'd start looking. In that order. Winchester makes several versions of the 70, but the Featherweight can be found right at your $800 budget. Most of the other versions will be over budget.
Forget iron sights. Most manufacturers stopped using them for good reasons. If you must have them you'll have better luck on the used market. I'll soon be 60, there haven't been any commercial rifles made with decent irons in my lifetime. Most are cheap, and most modern scopes do everything better. It ain't expensive to get a rifle drilled and tapped for irons if you buy one without them. Good aftermarket irons are available, but will cost more than a decent scope. Even if you buy a rifle with irons, I'd replace them with something better if I actually planned on using them. I just took mine off so they don't hang up on brush.
If performance and value are more important look at Tikka, Howa, Weatherby Vanguard, Savage, and others.
And since you already have a 308, stay with that caliber. Common 308 rounds easily out perform 30-06 loads from just a few years ago, and do it with about 25% less recoil. Of course modern 30-06 hand loads will still beat 308, but unless you're moose hunting at extended ranges no animal will ever know the difference.
