Usually, the leak that you describe is caused by a worn out flapper. Replacement is very easy and the part is cheap. But there are other things which might cause it.
Occasionally, the fill valve either needs adjustment or replacement. To check, take the lid off the tank. If the water level inside is below the top of the tube that comes up from the bottom, it isn't an adjustment.
Next, pull the thin, flexible tube that comes from the fill valve and goes to the tube, if there is water coming out of it, the fill valve needs to be "rebuilt" or replaced. Rebuilding sometimes works, but again, the part is cheap and replacement is not very difficult.
And the least likely, water could be coming through the seal at the bottom of the tank, below the flush valve. It is definitely possible to replace it, and less expensive than a new toilet.
Let us know what you find.
as to cleaning the bowl, bleach and scrubbing won't do it, but here's a sight which explains lots of tried and true methods.

http://homerepair.about.com/od/interiorhomerepair/ss/remv_rust_stain.htm