In order to support two monitors today, and the addition of a 3rd 4K monitor in the future----you'll need a video card which supports 3+ video ports and at least one of those ports must also support 4k output, i.e. 3840x2160 resolution, at a minimum.

I research/review products, especially all the details in the SPECS, at the manufacturers webpage, and that's usually after searching/comparing/contrasting similar products on sites like MicroCenter, NewEgg, Amazon, as well as the former FRYs Electronics.

The Radeon RX 560 Richard Shaw suggested, has 3 video-out ports (1xHDMI 2.0b; 1xDisplayPort 1.4; 1xDL-DVI-D) and supports resolutions up to 4096x2160 (check!).  Newegg has it for a price within your range.  I suggest reviewing the graphic card power supply requirements versus the rest of the equipment in your desktop (i.e. total up all the watt's consumed) and ensure your desktop power supply sufficiently exceeds that capacity.

I purchased a pair of 4K monitors early last year to use with my new desktop.  I choose the Samsung - UJ59.  Great 4k resolution at 32" in size, shop around for price as you should be able to secure one for $300-$400.

As for native open source drivers on Radeon equipment I cannot speak to that.  My preference is for something out of the nVidia family of products as their drivers for Linux (Fedora, Red Hat, Centos) have just worked whereever I've used them (at work and at home; physical systems and virtual).  One card to consider would be the GIGABYTE GeForce GTX 1050 TI N105TOC; it is very close to within your price range, has 3 video ports like the Radeon RX 560 card and outputs resolution up to 7680 x4320.

Thank you.

R,

-Joe Wulf