I would like a gun suitable for hunting deer and elk.
Well now, if its elk you're hunting, you may need a .06
Now your gonna spend money.
Sounds like you need an 30 .06 Partial to remington myself but you may need to by a used rifle with that budget. Save up for a little while so you don't buy one with a barrel thats shot out.
Invest in a Rem 700 in 30-06 then...I use a 30-30 Mod 64 Winchester for deer and a BAR 300 Win Mag for Elk,but the 30-06 makes for a good all around gun especially for the first time shopper.
Bolt action. Rem, or Win depending on which one you like. Distance and preference would determine if it's scoped, or not. Also, scoped rifles used around, or through the dew point can be useless. Depending on distance and preference again, 308 has range limits on Elk. 300 Win mag is good on all N. American game, including Elk.