Your first real hunting rifle should not recoil more than a 30-06 in my opinion. The 270 Winchester would probably be a better choice.
Hey OP, I strongly agree with the above statement by redneck09. I also started with my first own real hunting/sporting rifle on this forum. Some excellent advice from people here.
I bought a Howa 1500 in 30-06 based on discussions, and I'd say that the platform is a great choice, especially since for the price of the bare Tikka T3X you can get the Howa with a thoroughly decent optic. In terms of accuracy, I'm not sure you will notice any difference.
However, in terms of caliber, if you haven't done much shooting, a 30-06 will be quite thumpy. If this is your preferred cartridge (as it was for me), shoot 150 or 165 grain bullets, until you get familiar. 180 grainers beat up the shoulder.
Also, if you plan to do a lot of paper punching, the Thuddy-Aught-Six is a pretty expensive way of doing it... not that 270 will be much cheaper.
Finally, redneck suggests a Savage 110 as an alternative. I would disagree with this, if you insist on 30-06 or even a 270. I have a Savage 110 in .243, and I wouldn't want it in a heavier caliber, at least not in the configuration it's in. Also, the action on the Howa is much better (smoother, more solid) than the one on the Savage.