The standard for desktop monitors seems to be a matte (anti-glare) finish; if it's not specified for a particular model I'd assume that's the way the monitor is coated.
There's no required order to do this in, so it's up to preference. I might go for the GPU first so that you don't see any possible performance loss due to moving to a higher resolution on the integrated card in the period between the two purchases. Actually you'd see a big performance increase right away, and can then upgrade the monitor later and keep the performance boost.
Do any of you know which manufacturer seems to be best?
From what I have read, this MSI 660 Ti appears to be a good mix of stability/low temps/quiet. So far, it looks like the one I will be purchasing in 2 days (fri, sept. 5th), unless I can be persuaded otherwise.
@regor245 and whoever else may have been confused about my budget range I apologise, I should have made it clear that I meant ~$300 each. Thought I fixed that before I posted