Basically, the training of any of these nurses would be the same. After graduation, they may chose to take extra courses in their specialist area. For example, the LTC nurse might want to study gerontology.
The acute care nurse is your 'average' hospital nurse. Her role would vary a bit depending on which department she worked in, but hospital nurses in general provide hands on patient care to patients who have surgical or medical problems. They do dressings, IV's, vital signs, transcribe Dr's orders, update the charts, etc, etc.
Homecare nurses travel around the community seeing pts. who for the most part, have recently been released from hospital. As well as doing dressings, IV's, vitals, etc. they have to liase with the homecare system (in Canada, we are under contract to a government run body called the Community Care Access Centre). We inform the CCAC of what the pt. needs in the way of equipment and supplies, homemaking, physio. (Often these things have already been ordered, if the CCAC case manager is on the ball.) When necessary, we call the doctor to arrange tests or suggest changes in medication/ treatment. A hospital nurse would not have such a proactive relationship with the doctor...quite often, I've found myself telling the doctor exactly what to write in the way of an order!!
Yes, some days I looove my job!
LTC nurses have more of a supervisory role. Most of the hands on care is given by health care aids/PSW's. The nurse gives out meds, does treatments (dressings, etc.) transcribes doctors orders, and supervises other staff. I've worked relief assignments in LTC, and except for doing med rounds, a lot of the time is spent at the desk answering the phone and doing paperwork (charting, etc.) Depending on the institution, on night shift, the nurse might find herself helping to change and turn patients. Be prepared to change a LOT of diapers if you go into long term care! Good skin care is extremely important to bed and wheelchair bound patients.
A suggestion: why don't you hunt up a nurse who works in each of these fields, and interview her? The role of the nurse varies slightly from institution to institution, and what I've said about homecare may not apply to the United States. It will be a great way of finding out the pros and cons of each of the different kinds of nursing work!