Regarding what type of bulb - getting a good lighting effect is seldom a function of the bulb (leave color aside). In any reasonable working distance, all bulbs become a point source equivalent (*). When doing portraits, it is usually desirable to create a large light source, thus having "soft" light (note - soft, not flat). The way to achieve this is by using some kind of diffuser or soft box. An easy setup can be a shoot-through umbrella.
(*) an exception is a long fluorescent bulb, but still it is a 1 dimensional light which may create harder shadows in the perpendicular direction.
Update: Summary of the discussion in the comments, to OP's request:
What umbrella can I use?
A "shoot through" umbrella is the semitransparent white one. A reflector umbrella can be used as well. The idea is to enlarge the effective light source. The advantage of shoot-through is that the "source" can be located much closer to the subject (technically, to 0 distance) than the reflector (where you are limited by the shaft length). This means that compared to the subject, the relative source is bigger. The disadvantage is that being semitransparent, you reduce the power of the light source.
Can I use the lamp's "shade" instead?
I'd say that a "shade" is similar in effect (just smaller). You do want a white one, though.
Where can I find such an umbrella?
Here's one: dealextreme.com. You can easily find more on eBay or Amazon.
What should be the power of the bulb?
Generally, power is "the more the better", but be careful of overheating. A dimmer controlled light will be best to get the desired effect. Weaker light can be compensated by longer exposures. You want stronger light mainly because you want shorter exposures to minimize motion blur. Also, the stronger your source, the more dominant it is and allows you better control of you lighting setup. A dimmer, however, will let you decrease the power according to the actual need, and basically will allow you to get closer to the subject.
That said, a cheap off camera flash with manual mode is far more flexible than a fixed light fixture. You can find some at the same websites I pointed to above, together with an optional optical slaves (to be triggered by your on camera flash). Some have an integrated slave. For others, use a standalone slave.
This and this can be used as fixtures for (a) bulb(s) with umbrella.
OK, I am almost convinced to try an external flash. Can you recommend an affordable one?
YONGNUO is manufacturing a series of affordable flashes, some of them have manual mode and slave. As an example, the YN460 is a basic one with nice features. (don't stop at this one - do further research on other models they have and other brands. I remember the Strobist recommends a cheap manual flash as well, which was designed using his inputs so must be fairly suitable for a home setup).