Generally this day in age the price is indicative of the features or quality. Every now and then there are outliers to this rule but they are rare I think. (Ie a cheap rifle that shoots great or a reputable quality rifle that shoots bad).
For what you've described, a $600-1400 rifle would probably suite you best. 6.5, 7mm, or 30 caliber would do well. Lots of choices on cartridge. Sounds like you don't reload. Quality factory ammo is likely your route so I'd choose the cartridge accordingly. 30-06, 300 win, 300 WSM, 7 rem mag, 7wsm, 280, 280AI, 6.5x284, 6.5 RSAUM..... the list is endless and all would do the job well.
My personal recommendation would be 7 rem mag, then 300 win mag, then 280.
What people have said about barrel length needs some clarification to me. Barrel choices are a constant compromise decision. Heavy holds well, but it's heavy to carry. Fluting reduces weight but it's expensive. Longer length increases muzzle velocity but it adds length you have to maneuver.
But I understand what they're saying...in most cartridges 4-6 inches of barrel isn't going to make or break anything performance wise. It will still be accurate. It will still kill an animal. Then general rule of thumb is that 1 inch of barrel is worth 25fps MV.
The only other advice I have is that if this becomes a growing interest for you the Remington 700 is one of the most popular actions to build a custom off of.
Good luck