For informal target shooting and learning, the 30/06 is just fine. I have used a 30/06 in competition at 1000 yards and it worked quite well.
The 338 Lapua, while it is undoubtedly effective enough, is an over-rated cartridge which has been successfully sold to various governments. I have attended matches where a $5500 rifle in 338L has been no more than competitive with 6.5mm rifles costing 1/3 as much. I am not saying the 338 cannot shoot well. I am saying, if the goal is to score well on a target at 1000 yards, it is no better than a host of lesser cartridges. GD