Weatherby Vanguard or Howa 1500 are essentially the same gun priced new under $500 with synthetic stocks. I wouldn't even consider them budget guns, but first quality rifles, put a nice stock on them and you have pretty good quality. The downside is that they are the heaviest rifles made. I've had a few over the years. Great rifles, but overweight for my needs.
You can find Remington 700 ADL's new for under $500. I think the Vanguard/Howa is a slightly better rifle, but I'd rank the 700 above it just to get a lighter rifle.
I cannot suggest the Savage Axis, Ruger American or Mossberg rifles. Nothing wrong with them, but for the cost of 2-3 boxes of ammo difference there are just much better choices.
Your best bet is a first quality Ruger,Remington, or Winchester bought used. I've bought/sold/traded hundreds of guns in my lifetime and have had more problems buying new than used. Very few centerfire big game rifles are shot enough to come close to wearing them out. At least 80% of the guns I've owned in my life were bought used.