I used to work in the Financial sector, but not in any areas related to trading or derivatives. If you just want to build up your knowledge in general terms I would ensure you are familiar with: Basic Accounting and Financial Statements. Useful texts would be
Horngren: Accounting OR
Weygandt, Kimmel et al: Accounting Principles
Also learn the basics of investments and corporate finance
Bodie, Kane, Marcus: Investments AND
Brealey, Myers: Corporate Finance
This will give a good understanding of the language and theories of Accounting and Finance.
Now if you want to get into more the quantitative side of finance, there is no better introduction than that of
Wilmott: Paul Wilmott introduce Quantitative Finance
I have no knowledge of algorithmic trading so I can't recommend any books in this area. Programmers sometimes find work doing things such as derivative pricing, although my observation has been the it is often the guru 'quants' with PhDs who usually do the work in this area. Well known books include
Joshi: C++ Design Patterns and Derivative Pricing
Note also that quantiative finance will require a strong mathematics background.