Let me do what I do best - stirring the hornet's nest

First and foremost, let's look at the meaning of the word "programming". Since I can't show you my dictionary, I will refer you to answers.com. I will focus on the technology section and the first part of the 2nd definition,
"A person who designs the logic for and writes the lines of codes of a computer program"
By this definition, "programmers"- web developers and DBAs - are all programmers, by definition. However, by disclipline, is another matter. You could choose to be either a software (java, c, c++, etc) developer, web developer or DBA.
Sorry for getting on a soapbox. This, together with calling web developers "designers", has always been a pet peeve of mine. Now, let me answer your question

:
What you should pursue should be a combination of "what you love versus what you know you know versus what the market demands at a particular point in time". Certainly, if you go for, say, DBA because you know it well, yet the market has no demand for it, then you are obviously wasting your time, effort and money. On the other hand, if the market is demanding PHP developers and you have a love for programming (this is where "logic" comes into play

), yet you don't know anything about PHP going in, assuming the company wants someone who can hit the ground running, your day of reckoning is only a matter of time.
Assuming you are in Nigeria, if you ask me, I would say know enough of all 3 disciplines to be dangerous, but like the kung fu masters of old would say, specialize in one. The one I would choose, based on what I have seen/heard about the naija environment, would be web development. There is still a lot of ground to conquer in this field over there, a lot yet to be unveiled. For instance, we are yet to have a true ecommerce online payment/ecommerce solution. Since we have been banned by the likes of paypal et al, we resort to being ripped off by the likes of epayassist and epaygold. As I have said on this forum time without number, whoever comes up with this solution will become an instant billiionaire. Then again, it's just my opinion.
as always, my 2 cents