There are a couple of strange things here.
As of the time of writing this answer, the current version of QGIS is 3.14, or 3.10 for the LTR (long term release). There should definitely be more in the Raster menu in any QGIS version I know (since 3.4) than what your screenshot shows, including Raster / Conversion / Polygonize
which should do what you want.
The doc link you provided is to version 2.8, which is very old; the big orange link at the top of that page will take you to a more recent one. The key elements of the tutorial you're referring should continue to work from that old doc, but various options will have changed.
I would suggest
a) Trying with a new clean user profile, in case you have somehow mangled yours.
b) Reinstalling QGIS, in case your installation is corrupt
c) Making sure you are using the latest standard QGIS version or LTR (3.14 or 3.10 at time of writing)