That's a tight budget, but doable if you do some homework and spend the time hunting local shops...you may need to go used...nothing wrong with a good used gun.
I would also disagree with the comment about avoiding Savage rifles...I have bought 3 Savages (Model 93 in 17HMR, Model 11 in 7-08, and a Model 10 FLCP-K in 308) over the past couple years and all are shooters...unfortunately all of which are out of your budget if buying new...but the Model 11 in 7-08 I picked up used (still had the stickers on it and was in like new shape and had hardly been shot) for $375 with a new Nikon scope, sling and very nice soft case.
The Ruger American gets good reviews as does the new Remington 783 are both close to your budget, but without optics.
Don't discount the NEF Handi-rifle (I have one in 243 with a heavy barrel and it shoots very well) or the Rossi single-shot....we bought our daughter one with barrels in 243, 22 and 20g as her first rifle and paid under $300, but again without optics and it was a good shooter and a good first rifle for her to start on...she's since moved onto a Weatherby Vanguard (Howa 1500) in 243 in a youth stock.