There are tons of different apps out there that you could use. I assume that you know about the big boys: PhotoShop (raster/pixel-manipulation), Illustrator (vector) and CorelDraw (vectors, although the package also contains a package for rasters). Here are some other things to check out:
I do a fair amount of 2D graphics (mostly as textures for 3D images). For this, I use Paint Shop Pro. It does both vectors and rasters, and does most of what PhotoShop or Illustrator could do, but without the price tag. It’s not available in Taiwan so far as I know, but you can download a demo from the JASC site and buy it over the net if you like it.
Also, if you buy a Wacom graphics tablet, it will probably come with either PhotoShop Elements (a cut-down version of PS, probably not quite as good as Paint Shop Pro) or Painter Classic. Painter is the most famous natural-media (crayon, watercolor, etc.) app out there.
It really depends on what you mean by cartoons, though. If you mean animated cartoons, you should probably check out Flash, which is becoming a standard format for streaming Web cartoons, or Toon Boom Studio if you want to do more traditional animation. If you want to do still cartoons (i.e., comics), it’s actually pretty common for people to just draw their cartoons by hand, scan them into a computer then color them with PhotoShop or PSP or whatever. If you want to draw still cartoons directly into the computer, you just need a good graphics tablet and a decent 2D graphics app such as PSP or PS.
It’s also a good idea to check out some of the CGI magazines that Ch