Import the height map using Import images as planes.
Enter edit mode and subdivide the mesh like a 100 times.
Add a Displacement modifier.
Set the texture to use the height map. Set the color space to Non-Color (height maps should be treated as data, and have no "gamma" curves)
Don't worry if the displacement seems exaggerated for now.
Back in the displacement modifier, set the displacement axis to Normal and to use the UV map.
The height for the displacement can be conrolled moving the strength slider.
For a better level of detail add a subsurf modifier before the displacement:
(be careful not to set the subsurf modifer to a high number, 1 or 2 subdivision levels should be enough, otherwise you might end up with an unmanageable amount of vertices and your computer might slow down or crash)
Then assign the color texture image as the color for your material.
Why the funky edges on the surface? Your height map needs a bit of cleanup before doing all of this
Usually the import images as planes plugin is enabled by default. If it isn't, you'll have to enable it manually. In the top menu find Edit > Preferences. Then from menu navigate to the add-ons section and find Import-Export: Import Images as Planes.
To enable the add on check on the box to the left of the add on name:
For versions previous to 2.8 the settings can be found here: