Yeah, those white box Remingtons are pretty disliked. Like Bosk said, I'm fairly indifferent to what I run through at the range. Afterwards I go home, clean the weapon, and load it with good ammo. I typically buy Academy's cheapo brand (I think it's Monarch), and while I wouldn't take the stuff to a gunfight, I've had no problem target shooting with it.
The cheapo Russian ammo from Walmart is TullAmmo. Steel cases and pretty dirty. I still wouldn't worry too much about it for range shooting, though.