Bishops Puzzle: 

I hope you will forgive my algebraic notation, but I found it easier
to read (since we know every move is a bishop).

White bishops start at b1 and d1, 
Black bishops start at b5 and d5.

01. b1-c2
02. d5-a2
03. b5-c4
04. c2-a4
05. a2-b1
06. c4-d3
07. a4-b3
08. b3-d5
09. d3-b5 (board is symmetrical)
10. b1-d3
11. d1-b3
12. b3-a2
13. d3-c2
14. b5-a4
15. a2-c4
16. c2-b1
17. a4-d1
18. c4-b5

This was not the first solution that I found, but I liked it because
it was symmetrical.  Once you get to step 9 you are basically 
undoing all the steps you have done.

Alexander Doskey

Knights Puzzle: 

1. Nd2 Nb4
2. Nc4 Nc2
3. Na5 Na1
4. Nb2 Nc3
5. Nd3 Na2
6. Nc5 Nc1
7. Nc4 Na2
8. Na3 Nc3
9. Nd3 Nd1
10. Nb5 Nb3
11. Nb4 Nd2
12. Nd5 Nb1

Alexander Doskey solves a harder version with four Knights each!

White Knights start in a1 through d1,
Black Knights start in a5 through d5.

01. b1-d2
02. d1-b2
03. d5-c3
04. a1-c2
05. c2-b4
06. b5-d4
07. c3-b5
08. b4-d5
09. d2-b1
10. c1-a2
11. a2-b4
12. c5-b3
13. b3-c1
14. b2-a4
15. a4-c5
16. a5-c4
17. c4-b2
18. b2-d1
19. b1-d2
20. d2-c4
21. c4-a5
22. b5-a3
23. a3-b1
24. b1-d2
25. d1-b2
26. d5-c3
27. b4-d5
28. d4-c2
29. c2-a1
30. c3-b5
31. d2-b1
32. b2-d1

Solving the knights puzzle wasn't too difficult, but it did take me a
few tries.
What was time consuming (extensive enumeration) was trying to find the
shortest solution (something that looked elegant, not like just trial
and error).

Alexander Doskey

Is this any shorter?

Chess Quotes

" It is often supposed that, apart from their 'extraordinary powers of memory', expert players have phenomenal powers of calculation. The beginner believes that experts can calculate dozens of moves ahead and he will lose to them only because he cannot calculate ahead so far. Yet this is utter nonsense. From my own experience I can say that grandmasters do not do an inordinate amount of calculating. Tests (notably de Groot's experiments) supports me in this claim.
— David NORWOOD, Chess and Education