Trans is an umbrella term for people whose gender identity does not match the sex that was assigned to them at birth. For instance, doctors and parents might assign someone to be a girl when they are born (they usually do this on the basis of the newborn’s physical appearance one’s genitalia), but later in life this person might realize that they are a boy or that they are non-binary.
Non-binary people do not identify with the sex that was assigned to them at birth and have a gender identity that does not correspond to being (exclusively) male or female. They might identify with being both, or neither of these two, their gender identity might be fluid, or they might not have a gender identity at all.
Cis or cisgender is an umbrella term for everyone who is not trans.
Transitioning describes the process of accepting, affirming and making oneself feel at home in their gender identity. Transitioning has many aspects, including social, medical, or legal. Social transitioning for instance means telling people (family, friends, or doctors) that they are trans. Some trans people also transition medically: some take hormones and/or opt for surgeries. Some trans people do not go through any medical transition. Trans people are an extremely diverse group and as such, their needs in healthcare settings are also extremely heterogeneous.
Many trans people want access to legal gender recognition, which means changing their gender marker in their ID cards or passports so it matches their gender identity. This is not really a huge ask from the state, just consider how cis people never have to worry about their IDs matching who they are. In many European countries, trans people have to fulfil all or a combination of severely intrusive requirements to make this happen: be forcibly sterilized, receive a psychiatric diagnosis, take hormones, go through medical examinations, and/or get a divorce.