The current senior bioinformaticians of today, had no formal training in bioinformatics. In the 1970s and 1980s there was no field of bioinformatics, instead there were areas of biology that required substantial use of the computer. Crystallographers had to write code to transform x-ray diffraction data into molecular structure predictions; evolutionary biologists had to create databases to store protein and DNA sequences produced by the introduction of sequencing technology. Arguably this is where the field of bioinformatics began, and is why structure and sequence analysis used to be the majority.
Now, students entering the field have formal training, and their backgrounds can be roughly divided into two categories: computer scientists who enjoyed biology at school and are interested in tackling biological problems, and biologists who are handy with computers and fancy taking this from a hobby to a profession. There are scientists with backgrounds in statistics, maths, and physics, but in my experience these are in the minority compared to biologists and computer scientists.
The usual entry to the field is via a masters degree, which most universities offer. A masters in bioinformatics is conversion course that teaches computer scientists the required biology, and biologists the essential computing skills, such as programming. Since very few students have a strong understanding of both computers and biology, the aim these courses is to make sure they leave with a basic understanding of both, as well as familiarity with current topics in bioinformatics research. The structure of a bioinformatics masters is usually a combination of lectures and practicals, with a large research project.
It is debatable which type of background, biology or computing science, is more advantageous. Biologists have several years of biological knowledge and experience writing scientific papers. Computer scientists on the other hand, have the advantage of understanding all things informatics, which has a steep learning curve for the biologist who has never programmed before.
So once you have your master's degree, and you've decided that bioinformatics is what you'd like to spend your immediate future doing, what can you expect? Given the hype, bioinformatics is still a job like any other, where over a long enough period of time, day-to-day work can border on the mundane. Also, as a scientist the amount of time you'll spend writing code or producing results will decrease as you further your career. More and more time will be required for reading, reviewing, and writing grants and papers. If you eventually become a university professor you'll also have to manage your own research group which may or not be your idea of a good time.
Personally, what makes me stay in the field is how interesting the problems are, and the independence I have in solving them. When I find a question I'm interested in, I can use my initiative to answer it in the way I think best. I'll usually then discuss the results with my colleagues or supervisor to get their opinion. The best thing for me is that, after specialising in data mining and statistics, I can start to do things that other people can't, which means other people approach me to ask for my opinion on their problem - which is quite satisfying.
When people refer to the field of bioinformatics, they're referring to what might be aruged as two overlapping areas. The first is what you would call "bioinformatics", which is more technical, and examples are creating tools to analyse data for biologists, or specific databases to store and retrieve information. For example if you created a new tool that could analyse sequencing in a way that hasn't been done previously, then this is bioinformatics. Many journals such as Nature and Bioinformatics, have sections purely for articles about new methods and tools.
The second path is what you might call "computational biology", which is all about doing biological research, using a computer instead of a pipette. A strong understanding of biology is important, as well as the ability to phrase, then answer a research question. For example, if you believed that duplicate genes were less well conserved compared with non-duplicates, and you tested this hypothesis across a set of genomes, then this would be computational biology.
These two fields are not distinct, and overlap a fair amount. Some universities have bioinformatics departments in both the computer science and life science faculties, indicating the types of research carried out in each.
Salary is the most over hyped part of a bioinformatics career. You're never going to be earning six figures unless you're a senior professor or can do something that very few others can, which also has some commercial application. As an example, if you have developed a new method that can predict molecular interactions in the human body, this can prevent wasted research and save a drug company a fair amount of money. However, being able to parse BLAST reports for orthologs to a candidate gene, while undeniably useful, does not have commercial potential.
Overall, searching around on the Internet shows that, in industry, bioinformaticians are relatively poorly paid compared to other IT professionals, while relatively better paid compared to life scientists. So it depends on which of those two you prefer to judge yourself by.
On the other hand, if you're working in academia, then you'll be paid a similar amount to scientists at your level, e.g. PhD student, post-doc, or professor. However, if you do work in academia then you're probably not doing it to pay for your playboy lifestyle.
Industry vs Academia
This brings me to my last point, there are two different career paths for bioinformatics, working in academia or in industry.
In academia, the career path is something like PhD student, post-doc, fellow, principle investigator, and then professor. The numbers at each stage are dramatically reduced, and an indication of the attrition that takes place. The early part of a career in academia is judged on how many papers published, while later as you become a group leader, the amount of money brought in through applying for grants becomes equally important. An academic career is less secure, but you have the benefit of being able to work on whatever you want, as long as you're successful.
In industry, you have less freedom to work on what you choose, you work on what your employer wants, but to balance this you have greater job security. The commercial nature of industrial research means that publishing papers is secondary, or prohibited in the case of sensitive material, which can make the move from industry to academia some what difficult.
So, these are my thoughts on a bioinformatics career. Bear in mind this is completely subjective. If you like computers, biology, and solving problems, then bioinformatics is probably worth a go. Just try to be realistic about your expectations, and don't pay too much attention to the hype.