@Unlearn: Everyone should care for themself at least a little, but it is important to care for others as well.
It's great to hear that you definitely do have empathy for people. Most everyone I know is completely stone-cold when it comes to the well-being of others.
You do seem to genuinely care about this (at least a little bit, anyways) and I think that the more you try to concern yourself with the needs of others (it doesn't have to be strangers, it can just be with people you're close to if you'd like) the more you will get in the habit of wanting to help them.
Of course you don't have to give away everything you have or devote all of your time to assisting people, but practicing holding the door for people (if you already don't) and doing small tasks like these can help.