I've had zero frustrations with my two Savage rifles - 12 LRP in .260 and 10FP-SR in .308. Actually, nothing but praise for my Savage Mark II TR .22LR, too. My 12 LRP has been a dream from day one, nothing added but optics. The 10FP is lower cost yet still very accurate, but the stock is nothing to brag about. My 10FP-SR is getting an MDT chassis and Thunderbeast, so that's more of a project gun for me.
On the Remington side, +1 what Sig Marine said.
In general terms, Savage is (arguably) quite accurate right from the factory and easier to do your own gunsmithing on (re-barrel, etc), but there are more customization options for Remington and you can make a 700 positively kick ass with some extra benjamins. Classic Ford/Chevy Pepsi/Coke thing here - I've made my choice and I'm perfectly happy, but in the end the "best base rifle" for you will be one or the other.
Much more realistic than realizing factory rifles are pieces of crap for the most part and selling it at a loss. Or dumping a bunch of money into one and selling it at more of a loss.
You get a rifle that the only way you're going to want to sell it is to upgrade to a new model, because you're hard up for cash, or to get out of long range shooting. So why not just cut to the chase? It's cheaper in the long run, I wish I had just bought an AW when I got into this well over a decade ago. With all the upgrading and selling I did because there was always something wrong I could have three AI's for just what I lost.
An expensive pill to swallow, but in the long run this is generally true for most things. Buy once, cry once.
If money's tight, I think it's OK to get started with basic stuff, but plan to use it for years and run it to death while you save up for what would have been the best choice the first time. The worst plan is starting at the bottom and upgrading (at a loss) to the next slightly better level, over and over and over. Definitely the most expensive and painful way to go. I've done it before ... not smart.