Is Sanger sequencing still used in labs, and therefore worth learning?
Sanger sequencing is still used in the labs today - and not only on the side. Next-generation sequencing has its strength when it comes to sequencing very large amounts of DNA (basically whole genomes or exomes).

Sanger sequencing is used when you want to sequence smaller regions or portions of a genome/plasmid. Typical read length is (depeding on the machine and the skills of the operator) 600-900bp. This is enough to verify site directed mutagenesis, the presence of inserts you cloned as well that they are correct (if the insert is longer you will typically nest the single sequence reactions to cover the whole sequence) or look for a polymorphism. There are endless possibilities.

Additionally it is relatively cheap to sequence single samples (which is not really working on the next-gen machines), one sample comes between 8 and 10$ (at least the last time I sent this out). And you don't even have to do it yourself as there are a lot of service providers available. The technique itself is not hard to learn either.