The 7mm Remington Magnum is significantly hotter than the others. It will stretch your effective range (if you don't develop a flinch).
The other rounds are actually not all that different in power, etc.
For general purpose if you're not a reloader, I'd suggest a .30-06 for the variety of factory loads available (again, the .30-06 is at ROUGHLY the same power level as the .270, .308, and 7mm/08 cartridges).
I know there are differences among the cartridges listed, but except for the 7mm, there's not enough practical advantage in power, range, accuracy or trajectory to recommend one over the others.
Of the choices you provided, I'd probably pick the .308 for ammo versatility and availability, and the variety of BA rifles chambered for it.
All those cartridges are fine choices for an all-around rifle, unless your "all-around" shooting includes the big bears of North America or the Big 5 in Africa.