Talk about transgender people in the right way

Avoid the word "transsexual". Some trans people still use this word for themselves, and that's okay because they've reclaimed it, but you shouldn't go around throwing this word on people. A lot of trans people consider it an outdated and derogatory word. A transsexual is generally used to mean a transgender person who has genital surgery.
Don't say "transgenderism". That makes it sound like a disease. If you are trying to talk about the overall experience of being transgender, I think the best way to rephrase your sentence is simply to say "being trans" or "trans people". So, instead of saying,"there's been a rise in transgenderism," say,"there's been a rise in people coming out as trans". In most cases, I think it's best to just reframe your sentence to be more specific to what you are talking about.