Bioinformatics is all about research. That involves search algorithms, databases and programming. Math is very important. Bioinformatics involves theory about algorithms on strings. Building phylogenetic trees for species (quite a big subject). Two research groups in my university worked on. Why and how a specie of bacteria lose genes when adapting from one host to another (their genom become smaller). How histones worked (DNA packing molecules, involved in regulation). I think having a passion for math is good. Since the field is growing, if you have a phd you can quite easily get a research job! Getting there is another thing. Bioinformatics is one of the hotter research areas. I think my teachers in bioinformatics was very competent. It is also a very international environment. The problem was that our schedual was made up by the biology faculty and by people who never had programmed.
I have taken bioinformatics myself, unfortunately the program was too hard for me. I also could not see myself living in Stockholm (capital of sweden). SICP in two months, a lot of logic, in the first course! (I never quite figured out a good strategy for programming in scheme). The professor was mentioned in the SICP book though wich was pretty cool. The faculty also did research with supercomputers. If you want to make money you should probably reconsider.
Most people educated in bioinformatics work as developers, researchers or in statistics. In research you can choose between classifying cancer patients in hospitals (quite common) or by doing research in a university (if you are good). Some do molecule modeling, some do extremely specialized bioinformatics where there are themselves and perhaps 20 other people that read their reports. There is also research environments outside the universities that does bioinformatics exclusively. As a developer you most probably work in java or maybe with databases (if you are good, since database developers generally are paid more than other developers), some work in matlab and c++ as well. That is what I know about the field. England, Germany, USA and Japan have probably the most research in bioinformatics. Two links I found which are good.
Oxford - Computational Biology Research Group ,
Max Planck Institute - Protein Evolution