I suggest you do what I did and simply Google "shore fishing tenerife". The first page of results is full of fishing forums with people asking the very same question. There are loads of answers by people much more qualified in this area than me.
Regarding the types of fish found around the Tenerife shore, check out this page on my site.
At the moment, my site features information on freshwater and deep sea fishing in Tenerife, but nothing about shore fishing. As a result of your question, I might try adding this, so thanks for asking it!