I am looking for something that can be used to plan hikes and discover new areas.
Mapping is done by the federal states (Bundesländer), so you have to look at the respective land surveying office (Landesvermessungsamt).
E.g. for Bavaria, there is the Landesamt für Digitalisierung, Breitband und Vermessung Bayern, and they link to their own mapping site, the BayernAtlas.
You can find links to all Landesvermessungsämter at the end of this Wikipedia article.
An overview of the mapping sites (2013): Die Bundesländer und ihre Online-Kartendienste für topografische Karten.
I don't think there is something official similar in Germany. You can use Kompass, which use OSM data. I don't know where they get the elevation model from though.
As user24582 wrote, official maps are available in many places. The best freely available map data (available online, using smartphone apps or a (Garmin) GPS device) is based on Openstreetmap - crowdsourced maps. In most areas the data is by far better than other online maps, often even better than official maps which tend to be a bit outdated and not so much focused on tourists.
There are many OSM-based themed maps available, specializing on different use cases, such as hiking, car-routing or even horse riding.
For offline use (smartphone, GPS) my personal recommendation would be http://freizeitkarte-osm.de/ .
For online use I recommend https://hiking.waymarkedtrails.org/ which includes many hiking routes.
Here you can find a comparison of different OSM based maps with Google maps to get an impression of the level of detail available: http://tools.geofabrik.de/mc/#14/50.1096/8.6855&num=4&mt0=mapnik&mt1=google-map&mt2=public_transport&mt3=hike_bike