Good on you for wanting to make the effort. In my attempts to speak Spanish during my time in Tenerife, the only major difference between "mainland Spanish" and "Tenerife Spanish" seemed to be that, in Tenerife, they don't do the famous "Spanish lisp".
Thus, the word gracias, while being pronounced "grathias" in mainland Spain, would be pronounced "grassias" in Tenerife. It is also common to not pronounce the final "s" in words, so that example could equally be pronounced "grassia".
The only other major difference I encountered was the word for "potato", which in mainland Spain is patata, but, in Tenerife, papa.