whatever, what are you planning to do with the rifle? If you are just planning to go out to the range, shoot and hone your skills, then there were many good suggestions. If you plan to shoot competitions, that may narrow your choices a bit.
I've built a couple of rifles with the specific intention of shooting F Class with them. I went with a Remington 700 for a few reasons. From this perspective, I'd say that the first is that they are highly customizeable. Stocks, barrels, detachable magazine systems, triggers, etc.
If you are getting this rifle with the intention of getting into gaming, a few things will become important. They did for me, anyway. My considerations, in no particular order, were:
The stock. Everyone that makes a stock, makes one for the Remington 700 action. Finding one that meets your tastes will be easy.
The trigger. Again, just about everyone that makes an after market trigger, makes one for the Reminton 700. Jewel, Timney, Shilen, etc. I run a Jewel on mine.
Barrel. This isn't as big a deal. A competent gunsmith can fit a good barrel to just about anything, but finding a good gunsmith that can work on Remingtons isn't very hard.
Detachable magazines. This is the area that poses a bit of a problem when going with something other than brands that are widely used. There are many detachable magazine systems available for the Remington 700 action which use the widely accepted Accuracy International 5 and 10 round mags. Having a high mag capacity is advantageous when you are shooting a match and want to take advantage of a favorable wind condition. In many instances, going with a rifle that forces you to stick with some sort of proprietary detachable magazine will leave you with very limited capacity and having to spend a lot of money on spare mags.
Although I'll admit that an out of the box Remington is not as refined as a Tikka, I think that the Reminton will take you further down the road as you get more serious about shooting long range.
If I was looking for a rifle with a $600 budget, I'd be looking for something like a used SPS Tactical or maybe a Savage 10FP. Everything that I said about the Remington, pretty much goes for the Savage, just a bit less so. Put a good trigger on it and some good glass and get shooting. I wouldn't worry about the other stuff until the need arises.
As far as the glass goes, quality beats quantity. For tactical types of shooting a 3-15X should do you. I run a 5.5-22X50 Nightforce on my rigs and really like them.