The 308 Winchester is a cartridge. The 30-06 Springfield is also a cartridge. A cartridge consists of a brass case of specific size and shape, a primer, powder and a
bullet. Both of these cartridges use a 30 caliber or .308 bullets. Note the first two numbers of the cartridge name, 308 and 30-06, 30 caliber. As is the 300 Win Mag, 300 WSM, 30-30, 300 Weatherby Magnum , so on and so on.
So, what bullet? Pick one,
http://www.midwayusa.com/browse/BrowseProducts.aspx?&tabId=19&categoryId=1 7540&categoryString=652***19785***9016***7243* **&brandId=1165
If you are going to hunt with these bullets do not use FMJ or full metal jackets. If you want one bullet for both cartridges I would go with a 165gr Interlock.
http://www.midwayusa.com/viewProduct/?productNumber=416036
If you are going to get two different bullets I would go 150gr for the 308 Winchester and 180gr for the 30-06.