I am often asked what fish are good for beginners, and that can be a tough question to answer, since there are a lot of variables like tank size, and whatnot.  Goldfish tend to be the common starter fish, but an understand of the physiology of goldfish will lead you to realize that they are best left for large ponds, in the long run.
 
Below are a list of fish that are commonly available, typically healthy and are robust enough for beginners:

• Schooling Fish
  • Zebra Danios
  • Serpae tetras
  • Harlequin Rasobaras
• Non-Schooling fish
  • Gold gourami
  • Pearl Gourami

Some fish to stay away from:

• Oscars
• Pacu
• arowana
• any kind of knife fish
• African cichlids

I am not trying to infer that there is any particular magic around the recommended fish above, or that the others are "bad", just that certain fish are more likely to be succesfully kept by a beginner, without becoming a long term nightmare to maintain.  Fish like dwarf gourami's would seem to be a perfect fit for a beginner - they do not need much space, they are mostly very peaceful, and beautiful to boot.  But, the reality is that population of dwarf gouramis are so genetically weak that they nearly always die after a short time.