Finding rivers from GeoTIFF files with GDAL

As @Hornbydd suggested for rivers it is way better to download existing dataset. E.g. from hydrosheds or OSM (waterways and water layers). And clip that dataset to your study area.

You can relatively easily extract drainage. E.g. with Grass tools as suggested in other answer. If that is all you need, good.

But drainage is far from rivers, depending on detail/resolution you require, rivers are more likely areas (polygons). Getting a nice river network with perennial streams only, and excluding other drainage network, is mission impossible with SRTM data only.

Also for rendering purpose, you can use some satellite imagery. E.g. list of some sources on EOS website. You do not describe your 3D rendering technique. In open source GIS world you can use QGIS software to create 3D rendering with SRTM raster, and drape almost any GIS data (including OSM data mentioned above) in various formats as layers on it including opacity and even 3D objects.

QGIS has even an option to use OSM data as service - in multiple styles e.g. via Quick Map Services plugin. Everything already styled, all you need is load it as layer, load your SRTM and make 3D view of it.