There aren't really any set transition points apart from teenager, which strictly applies to all of the ages ending in 'teen' so from thirteen to nineteen. Puberty has nothing to do with it.
I'd say a child becomes a toddler when it starts to walk, before that it is a baby. Then child from about 3 or 4 years old (kid is slang) but that goes on to overlap with young teenagers. I'd say a 14 year old is both a child and a teenager. Young adult - this gets used to describe maybe 16-22 ish year olds. Adults - in theory everybody over 18, but generally the older teenagers aren't quite seen as full adults. Middle age - that's debateable! No-one likes being called middle aged so it is often 'your age+ another ten years' ![Smile [:)]](/emoticons/emotion-1.gif)
. But personally, I'd say roughly age 45-60. Words for 'older people' are quite difficult as some of them are seen as offensive in some way (you see how I used 'older people' - that is the UK's PC phrase now for people over 60). Senior citizens is another option (sounds a bit more American to me). 'Elderly' is a word you will hear a lot but older people hate it as it includes ideas of frailness and helplessness.
Adolescent - going through puberty (roughly) so tends to coincide a lot with teenager. Probably 12-16 or so. As soon as they start changing from 'kids' they are adolescents. Grown up - this is just a child's word for adult.