Heres a short guide about which language to use regarding trans people.
I will sometimes mention history but won't go into it deeply.

Transexual:
Originally there was the term transsexual - but the problem is that it implies it is a sexuality and I mean in the beginning it was seen as another kind of homosexual but regardless, that is a big reason why it was changed to transgender.

Today mainly 4 groups still use "transsexual":

• Old people who grew up with it and learned to use it (this can also be trans people who identify themselves that way).
• Generally people who just don't know much about trans folks and the language around it.
• Transphobic people who make this implication with sexuality partly on purpose, feeding into the narrative of "they are sexual predators" and so on.
  And they are generally a bit behind with science, so it makes sense that they use outdated terms/ reject modern language.
  They can also be trans people themselves: there is a group called "trans medicalists". These are basically mostly (white) binary trans people, who insist on all this medical diagnosis stuff and the pathologising of trans peoole, and want to define the term "trans" very strictly and often exclude non binary people or trans people with little/no physical dysphoria.
• Trans people who reclaim the term
  You have to consider that the term is very strongly medicalized. It was originally the diagnosis and was then replaced with disphoria and then with "gender incongruence" but amounts to roughly the same thing, with the latter having more inclusive definitions.
  They essentially just looked for prettier terms, so they don't say that being trans is a disease. However I was stilled called transsexual by 3 different medical practitioners, so the term is still used a lot because a lot of doctors don't have a clue about the topic.
  So some reclaim it for themselves after they experienced a lot of discrimination in the health care system because of the term. Therefore it has a sort of punk/rebellious tone like "queer" (the punk band "against me" with trans singer also uses the term)
  The term is also used as a noun often. As in "I'm a transsexual".

Trans identified (or "transident" in german):
Not good.
The problem is that it is rather a term that comes from cis people (especially transphobic ones) so they can say something like "trans identifying man" when they mean a trans woman.
It is not used in a transphobic manner in all contexts but should be avoided in general.

Lastly trans and transgender are adjectives which means it's not "transgendered" or "cisgendered" as that describes more something that would happen to you rather what you are.
It also means it is written "trans man" and not "transman" etc.

In the end you should stick to terms that are actually used by the community you are describing and there you are safest with transgender or simply trans.
(and if the person says it explicitly you can also use transsexual for someone)

Also for germans:
transgeschlechtlich: is basically the translation to transgender, but transgender is also used in german and is ok and actually better because gender just doesn't translate well into german.

Other Trans related terms

Transfem/ Transmasc:
Generally it's the umbrella term for people who transition from a feminen gender/ presentation towards a masculine or vise versa.
A transmasc person may be a binary transman or a non binary person who's afab and masculine presenting to some extent.

You might find exceptions like for example if you have a non binary person whos amab and presents more masc they might still relate more to the term transmasc than transfem because they are trans and masculine presenting. This usage is less common though.

Extra Notes:
Be sure to spell it "transfem" and not "transfemme" because "femme" with the extra "me" is a lesbian term. (unless you want to refer to a lesbian transfem).
Unlike trans or transgender, transfem and transmasc can be used as nouns. So you will hear people say "I'm a transfem" etc.

Transvestite:
Doesn't or shouldn't be used to refer to trans people as it's an old word for crossdressers.

Other Terms to avoid

Tranny:
This is just a slur so don't use it. Some trans people may use it for themselve sometimes however.

Ladyboy, Shemale:
These are terms from the porn industry to fetishize trans women so don't use them.

femboy:
Nowadays mainly applies towards just feminine boys or men in which case it's fine to use but historically it has been used against trans women a lot so don't do that.

futa:
This ones a bit more complicated and I'm no expert on it but generally nowadays it's a fetish term from hentai refering to women with a dick. Often but not always the characters may also have a fully functioning vagina as well as a penis, in which case it's also bad representation of intersex people.
Basically, many trans women and intersex people will find the term offensive if used in real life.
The porn category itself and the way the characters are portrayed is also seen as problematic by many but that's a different issue.