The '06 is one of my favorite calibers, and I've probably shot more rounds of '06 than all the others combined. I found that Sierra's 125 gr was very accurate, as was the Speer 130, but the Speer was very distructive. I never had any luck with 110's or for that matter with Remington's 55 gr Accelerator, which resulted in a bad experience when I was called upon to euthanize a poor old dog.
Cheap 150 gr Hornady flat base spire points did yeoman's service for many years, then I convinced myself that shooting 165's would give better results. The 165 gr Hornady boat-tail and the Sierra 165 gr hollow points both worked quite well, but once available the Nosler Ballistic-tips quickly became my favorite in both 150 and 165 gr.
I did quite a bit of shooting with 168 and 200 gr Match Kings, and satisfied myself that the old '06 can hold it's own on the target range despite it's detractors. It was also with these bullets in an '06 that I taught myself about long range shooting.
Then there is the X bullets, and I really tried to like them, but had trouble getting the early ones to shoot well - particularly the 200 gr. I decided on 180 gr. XLC's for bear loads in my wife's rifle, and with the luck I've had with TSX's in my .375 I'll have to try some in the '06.
I really enjoyed shooting cast bullets, and I did quite well with Lyman # 311284 which cast to about 210 grs, and chronographed at 1800 with a load of 25 grs of SR-4759. This proved to be a very good small game load.
I haven't had much luck accuracy wise with jacketed bullets heavier than 200 grs. in the '06. To some degree I guess I adhear to the idea that if I can't do it with a 180 from an '06 it's time to go bigger, but now that I've heard about Woodleigh's .30 caliber 240 gr. Weldcore, I am tempted to try them.
I've found the 06 to be very forgiving with extruded powders, and I've used the old standbys 4064 and 4895, but I've had lots of success with 4831 as well. I haven't had real good luck with Winchester ball powders in the '06, and it pretty much turned me off ball powder in general for all but handgun and straight-wall rifle cartridges.