Generating regular expressions not containing abc
Design a DFA that recognizes "abc" (make sure to include all transitions). Complement it (make accepting states non-accepting, and make non-accepting states accepting) in order to get a DFA for $L = \Sigma^* \setminus \{ abc \}$. Finally, write down the regular expression of the complemented DFA.



As a brute force solution that does not use DFAs: write a regular expression for all words of lengths 0, 1, 2, and more than 3. This should be easy. 
Then write, a regular expression for all words of length 3 except abc.
For example:
$(b+c)(a+b+c)^2 + ab(a+b) + a(a+c)(a+b+c)$

Take the union of all the above REs.