Any of the calibers listed are good choices,and perform well at either shooting targets or hunting.
I really like shooting the 260 Remington,while factory ammo can be hard to find and expensive,Handloading for it is the best bet. I make all my 260 brass by resizing 243 Winchester brass. The ballistics on the 6.5mm bullets are really good. I also just built a 6.5 Creedmoor,the case is a little shorter than the 260 Remington,but it uses the same powder charges as the 260,and you can seat the longer match style bullets without loosing the ability to use the rifles magazine because of an overall length being too long to fit.
The 7mm-08 is a great cartridge,and I enjoy shooting mine. I'm still trying to find the perfect load for it,so far it's been picky about bullet weights. It seems to like bullets in the 150-160 gr range.
My 308 is a great gun,it shoots just about anything I load for it from 155gr-210gr.
Like Axxe said,if you don't reload,either the 308 or 7mm-08 will be the best choice.Ammo is available almost every place that sells ammo usually.