The first flies I learned to tie were a San Juan Worm and Woolly Bugger. Later that day the guy showed me how to tie a Bead Head Partridge and Herl.
When I help with the fly tying at Scout Camp, I teach the San Juan Worm, Woolly Bugger or Woolly Worm, and a Foam Ant. All are simple to tie and all catch fish.
For great tutorials on a good selection of flies, there is this site and Charlie's Fly Box. Another site with really good tutorials is Eric Austin's traditionalflies.com Most of Eric's flies do not qualify as "simple" but Eric has a good tutorial on the whip finish and the techniques he shows apply to all kinds of flies.
The main thing is to have FUN.
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