I heard that programming and other computer-related jobs are being outsourced to places like India and China. Is this something to worry about and in this case, should I still try my chances at getting a job in the computer field?

To some degree, yes, companies are trying to outsource and ship jobs overseas. Quite often this is not all that successful, because the kinds of companies these jobs are shipped to (to save money) don't exactly hire the top-tier programmers (like here, the good programmers over there are working at better companies and getting paid better). As a result, quite a lot of companies are scrapping plans to outsource. In general, you may find entry-level positions hard to come by, but once you get past that hurdle, its generally not that hard to find good work (assuming that you are a skilled developer).

I agree with Rycross. Most of the work thats outsourced is about implementing than designing which doesnt require great software development skills. Most of the outsourcing that happens is in the services industry where its more efficient for a company to outsource. Product development is rarely outsourced as that requires strong designing skills and one would risk it giving it to someone else. Usually they outsource peripheral things like support or aging softwares.