Aside from buying an "off-brand" that most haven't heard of, you really can't go wrong. I've had my first deer rifle for 22 years, and I don't think I've put more than 6 boxes of shells thru it. Most from sighting in before the season.
Between your 2 calibers, I would go with the 30-06. Great all-around caliber and tons of ammo available anywhere. But, if you chose the 270, you wouldn't be making a bad decision.
I have a Ruger M77mkII in .308, which is basically a 30-06 in a short cartridge. My knock on the Ruger is the trigger pull, 7 million pounds!!! But, it's all I had for many years and it's shot it's fair share of deer. (ie: you get used to things). I picked up a Thompson Center Venture a couple years ago in 7mm mag. It's a fantastic gun, in your price range. Adjustable trigger and guaranteed MOA, and mine shoots fantastic. Has a composite stock that fit me better than its competitors, and I topped it with a Zeiss Conquest scope. As someone has already stated, spend your money on your scope. If you asked whether you should get a $500 gun and $150 scope, or a $300 gun and a $350 scope, I would suggest the latter. Go to the shop and shoulder a few. What feels good to others may not to you. Buy the one that 'talks' to you.