Hit walmart or wherever, and go to the waxes section. They sell little clay bar kits for about $15 for the meguiars anyway. It includes quik detailer spray, clay bars, and cleaner wax.
The clay bar shears dirt, grime, whatever off your paint and leaves it really smooth. After washing & drying your paint, spray on some of the quik detailer for lubricant and just rub the surface of the paint with the clay bar. it will get dirty and squish out, just fold over the dirty area and proceed. Basically you lube up your paint and scrub it with a chunk of clay...
As for really old bug removal, it may not do the job. My wife's car has bugs on it that I can't get off. But the claybar really makes a nice surface prior to waxing. Make sure you've got time to wax as I believe it removes the wax too.
Clay instructional video:
http://www.youtube.com/watch?v=yfEfLGL59GI