Well if you are just starting out, I would say you should look into a 30-06. It is a great all around caliber that's just good enough for just about everything. I am partial to it, but would be lying if I did not say that a .270 is not a sweet rifle as well. The ought 6 is a bit more versatile for larger game like Moose, and Elk, however a .270 in 130 grains is a rocket for antelope and sheep. Both have strengths and weaknesses, but are capable.
Being that you are new, if you don't intend to hunt Black Bear, a .308 will work well too, and costs a lot less to shoot at the range.
Now if you are looking for a rifle and a scope, for 400, you will probably not be able to buy a new gun; at least not a mainstream one. However, most used rifles like a Remington 700, or Winchester 70 are almost always found with low miles. People don't really shoot hunting rifles much. Usually they zero the gun with a few boxes, then shoot a couple shots a year. It is not uncommon to find a nice 20 year old rifle with less than 200 through it. I would look into a Remington 700. There are lots of good brands out there, but a 700 is the most common, and can be repaired cheaply.
I have never owned a Savage, and don't like them, but know hunters who swear by them as a "best bang for the buck" type of gun.
I will saw that I own several H&R rifles. These are single shot rifles that shoot great. You could get a nice 30-06 for about 275 dollars new. For 100 dollars more you could get a different caliber barrel for it, and have one gun 2 calibers for under 400. This solves the whole "which caliber do I get" thing.
What ever you do, try to spend a bit more on the glass. It is more important to buy a cheap rifle with a good scope than the other way around.
Any gun store can get you an H&R. They are also known as NEF guns.