There are typically two different licenses obtained by those working in the funeral industry: 1) morticians license and 2) funeral directors license. Most funeral homes prefer to hire folks who have/are working toward both as they are more versatile. This is also one of the remaining trades where an apprenticeship is required (usually 2 years).
Check online for the Funeral Directors Association in your state. They will be able to tell you the educational and apprenticeship requirements specific to your area. It would be ok to ask them, appropriately, what you essentially asked above (leaving out the fascinated with death part as indicated wisely by @poisonedantidote). I would just indicate an interest in the field but state that you aren’t sure if you can handle it or not and what do they recommend to find out if this is the career for you.
There are also a number of decently written books that might interest you:
The Undertaking: Life Stories from the Dismal Trade is very well written and sensitive on the topic.
Stiff by Mary Roach is an interesting book about folks who “donate their bodies to science” and is full of information about life after dying.
American Book of the Dead (much like its Tibetan predecessor) talks about the spiritual side of things in an important way in the current American funeral industry.
The Final Bath is written by a young funeral director and talks about the challenges of being a female in this very male dominated field. This one talks of her life working in a morgue. Her other book, Into the Hands of Strangers tells much more about the funeral industry from a woman’s point of view.