Does the degeneracy of the genetic code consist of two parts: 1. wobble -- the potential of a single tRNA to bind to two or more codons. 2. each amino acid has more than one codon that codes for it. Or is wobble the sole source of the variability?
What you have called (2) is degeneracy -- more than one codon for (most) amino acids. Wobble is a way of dealing with degeneracy -- it takes advantage of degeneracy to cut down on the number of tRNA's needed for accurate protein synthesis. I'm not sure what you mean by "source of variability." There is variability in base pairing (relative to standard G-C and A-U base pairing) allowed by wobble. There is variability in the codons that code for the same amino acid due to degeneracy. If this issue is still unclear, please ask again for clarification.

It seems to be that degeneracy and wobble means exactly the same things. From what I understand, degeneracy means different codons for same amino acid, and wobble means different codons for same tRNA. Then, since each tRNA carries a specific amino acid, don't the two terms just mean the same thing???
Degeneracy = multiple codons per amino acid. Wobble means one tRNA can pair with more than one codon (because of additional pairing options in the 3' end of the codon). You can (in theory) have degeneracy without wobble -- see problem set 7-22. You can't have wobble (more than one codon read by the same tRNA) if there is no degeneracy (without messing up the code).

Regarding Wobble- If the same tRNA can read more than 1 codon, i.e. GGG and GGA, then which anticodon does the tRNA contain?? Will it be 3' CCC 5' or 3' CCU 5'?
I think the exact rules depend on the organism. The usual ones are in Becker, table 18-2. U in tRNA can pair with A or G.