I'd stay far away from the 710's, or 770's. That said, I do know people that have them and for shooting the occasional deer, they do work. I really just don't understand why you'd want one of these though when you could get a rifle that's so much better for a similar price from another manufacturer. With the 770, you get a lousy trigger, a rough action, a barrel that's pressed in so if it wears out, it's too bad, and really an overall just cheap gun. Plus as said many shops won't take them on trade in, and the resale value is just horrible. A friend tried to sell his 710, and after trying for quite a while he finally gave up and just kept it.
For right under $400 you could get a Savage model 10/110 package at Walmart that includes the accu trigger, which is so much better, is a quality rifle all the way around, will most likely shoot sub moa groups out of the box, has a barrel nut, so you can easily change the barrels at home, and overall is just a rifle that's 100x nicer for not much more money. You will also get a lot better resale with this than the 770. IMO this is the way to go, however if you don't like this choice for some reason there are several other choices.
For $300 or under you should be able to find a Stevens 200. It's essentially the older Savage Model 110, without the accu trigger. It's also a nice rifle, and has many of the same traits as the above Model 10/110 it it just doesn't have as nice of a trigger, and to me it's worth the extra $100 to get a scope you can sell and the accu trigger.
Another option is the Marlin XL/XS-7. These can also be found around the $300 mark and are also a much better rifle. They have a trigger that is basically a copy of the accu trigger. Overall, these got great reviews for the early ones, but the ones with REM stamped on the barrel have been getting less than stellar reviews which is the only thing that makes me hesitate from really recommending this one.
Also for around $400 you can get a Remington 700 ADL at Walmart with a cheap scope. If you are dead set on a Remington, this would be the way to go over the 770, as the rifle is that much better. Although I think the Savage is a much better option.