I think 30-06 is my preferred caliber but have heard about accuracy inconsistency with different loads. While 270 sounds like a more consistent caliber it may not be powerful enough.
I actually recently just got my first hunting rifle and i also happen to be left handed. it is a lefty tikka t3 lite stainless in 308. ive shot it twice now putting a total of about 40 rds through it so it is still very new to me but it is extremely accurate. tikka sends a garuntee for sub moa with the guns and ive gotten right around 1in 3 shot groups at 100yds with mine and my limited shooting experience. if you can find one id say got for it you wont regret it.
as far as caliber 30-06 is extremely popular as it use to be the US military standard round and therefore has had rifles build in it for 100+ years. i personally went with the 308 because it will shoot the nato 7.62 surplus rounds that have replaced the 30-06 for military use. either way i dont think you could go wrong with either of those. ive heard good things about the other calibers discussed so really it comes down to what you feel you can find ammo for and suites your needs