I went thru this about 15 years ago. It's a fun but more expesive than it looks at first glance. I used a Interarms MarK X action(Large Ring Mauser), added a Shilen barrel and nice semi-custom wood stock for my 338-06. I couldn't be happier with the result.
Finding a good action will probably be the hardest part.
As far as caliber, if you're going to the trouble and expense of doing a custom rifle, get something that isn't found in every gun rack in the the country.