Answered your post on the H&R/Nef in the firearms review section.
That being said, there several great reasonably priced options out there right now. My favorite option is the Savage Model 10 they sell at Walmart or Model 110 if you want a long action. This comes with a Simmons scope, or some of them I've seen have a Bushnell new, and a sling. This rifle also has the famous accu trigger which is great. It's also very easy to change the barrels on these should you ever want to rebarrel. This is just an all around great rifle that will last a lifetime, as well as be very accurate with one of the best factory triggers you can get. The scope's aren't great, but should last for a while until you can upgrade. These sell for $387 at our closest Walmart and $397 at one about an hour away. In any case every time I've seen them they have been under $400. Really a great buy.
I'd stay away from the Savage Edge, not because they are bad, but because the above Savage Model 10 is a ton better gun and not much extra money.
Another good option is the Marlin XL or XS 7 depending on what caliber you get. These run in the $300 range without a scope and are also quality guns. You should handle these and the Savage and see which you like best. The Savage fits me better, but it may be the total opposite for someone else. These also have a similar trigger to the accu trigger. These are also known for being very accurate guns. I do want to add that I've never owned or shot a Marlin XL or XS 7, but I've handled several and read lots about them.
Then another good option is the Stevens 200. This rifle is basically the old style Savage Model 110. These are a good solid no frills rifle that will last a life time. That being said, for the $300 these seem to normally sell for I think the current model Savage is a better buy for less than $100 more with a scope and the accu trigger. The Stevens 200 has gone on sale for $179 at Academy Sports in 7mm-08 and I think .270 a few times. This is a great price if you can pick on up for this, I'd do it.
I'd stay away from the NEF single shots for the reasons listed in that other thread. Not to mention, who really wants to hunt with a single shot? With a Nef, the kids will probably outgrow it very quickly where these other rifles are something that they can use for quite a while and not really outgrow.
The other rifle I'd stay away from is the Remington 710/715/770 series. They are pretty much junk.
You can also get some great deals on scopes on clearance right now if you get a rifle without a scope and need to scope it.
As far as caliber, I think for kids .243 is a great choice.