I'll start by saying, I'm no expert, but either of those calibers would do great on the selection of game you mentioned. There are in fact several others that would also. For example, I'm a fan of the .270 win. However, with today's bullet technology, it's not the same debate it once was. Considering the bullet expansion control, weight retention and ballistics consistency, the major factor in deciding whether you would adequately harvest the game of your choice is actually the one with the finger on the trigger. Bullet placement, trigger control, reading a reticle, breathing and choosing not to over reach a distance target are what I would ask a new shooter to focus on.
That being said, buy the most reliable gun you can afford, in a caliber that will allow you to practice, practice, practice. The price between 30-06 and 308 is negligible since most hunters would use less than half a box per season. The price is important when you are a new shooter and need to practice, practice, practice.
Don't forget to budget for a scope. Don't spend all the money and effort in the rifle. Even the cheapest rifles would easily allow you to harvest game. As long as it shoots straight and bullet go where to want it to.
Most would recommend getting the most expensive low power scope you can afford. Although, I really like my Bushnell Engage. It's not all that $$$ but tracks excellent and is very clear.
The short answer is, no. The tikka is a great rifle and don't worry too much about the caliber.