Hi, I've namechanged for this because I'm far to easily searchable under my other posts 
I am a celebrant and have been doing this work for about 6 years. It is lovely, lovely work - flexible, creative, interactive.... but it can also be sad, upsetting and lonely. You need to have a good support network (and to be personally resilient) to deal with things like the funeral for a baby, or a suicide.
Depending on where you live, the type of ceremony you can hold will vary. Anyone can hold funerals and baby namings, but only certain groups can hold legal weddings. In England a celebrant cannot hold a legal wedding, and the market for non-legal weddings is relatively small and quite niche.
In addition to the range of skills you need to work as a celebrant they key to actually getting any work is to be great at marketing yourself and your skills. You will find that once you've trained you are pretty much on your own with this, so if knocking on the doors of funeral directors and selling yourself isn't your thing, then you might struggle to get work.
There is no nationally recognised regulatory body for celebrants, and there are many different organisations that offer this type of training. I suggest starting with a thorough look at what the different courses offer, how long they are, what they cost etc. As an example it might look attractive to have distance learning, or a course that costs £500, but think really hard about how credible it is, what you will learn and how supported you will feel. By comparison it costs nearly £4000 to train as a humanist celebrant for both funerals and (non-legal) weddings. Some of the celebrant training organisations are just that - organisations that make their money from training people, with little residual interest in what you do subsequently.
What are your goals? To be a successful funeral celebrant you need to be available flexibly during the week (when most funerals are held). If you are currently a SAHM with dc at school and few fixed commitments, starting in this field can be gradual until you build up a good reputation. However if you are currently employed or don't have access to flexible childcare, then it's going to be more tricky.
What are you expectations of income? I am really fortunate in the area where I work and I hold many ceremonies. However you'll not make a fortune, and I also do other work which makes up the bigger portion of my income.
Very happy to answer more questions if you want to pm me.