Transgénero is considered the more correct term now in Spanish, but transexual is still far more common, both in conversation and in the press. It's also becoming more and more common to talk about, for example, una mujer trans, which is a phrase that wouldn't have been used a few years ago.
We've had a big debate this year over the recently-passed ley trans (trans law), which will allow self-ID for people over the age of 16. Children of 12 will be able to change their legal sex with judicial approval.
Spain has, in the past, had some unnecessarily tit laws that have affected trans people. For example, legally changing your name is a very long and convoluted process (a friend changed her surname to replace a spanish ñ with a catalan ny...it took two years), and in Spain we have to use our legal names and show ID a LOT. So someone presenting as María with an ID card in the name of Jorge would have an incredibly hard time. That law was long overdue a change...but letting 12-year-olds change their documented sex? No.
(I just checked my ID and yes, we record sexo, not género.)