In QGIS you can use the 'Point sampling tool' plugin, or in ArcGIS you can use 'Extract values to points' from the spatial analyst tool box.
For both methods you will need to open the elevation raster and the point file with locations in the GIS platform of your choosing and make sure that both the elevation raster and the point file have the same projection.
More info on the point sampling tool in QGIS can be found in the following tutorial - http://www.qgistutorials.com/en/docs/sampling_raster_data.html
For Extracting values to points in ArcGIS have a look at - http://help.arcgis.com/En/Arcgisdesktop/10.0/Help/index.html#/Extract_Values_to_Points/009z0000002t000000/
It is very easy if you use a radar imagery in Global Mapper, and previously, if you have no any measurement points, you have to take points in google earth with "add Path", and after only using the TCX Converter free program, you can extract the real altitude of each point, and calculate the digital elevation model with QGIs or Global Mapper or ArcGis.
In ArcGIS, you can simply use the Extract Values to Points tool (ArcToolBox: use the search tool to find where this tool is): set it up with the input points feature class/.shp file and the input elevation raster, and it will add a field to the vector data and fill it with the elevation values at those points. If you only have a list of coordinates of the locations you're interested in, I suggest creating a feature class of points at those coordinates first, then using the tool as described above. Its pretty 'slick!