What is the total number of possible chess moves at White's second turn? At White's first turn there are $20$ possible moves: each pawn can move forward one or two spaces, or a knight can jump over the pawns to one of two positions each. Some moves are of course likelier than others (for example, in all the novice games I've seen, I've never seen anyone open with the king's pawn). Black also has $20$ possible moves at this stage.

At White's second turn, there is a small number of available moves based on what the first move was, plus a good player should take Black's first move into account. But what I want to know is, if we follow each of the $20$ scenarios created by the first move, count the number of available moves in each, no matter how strategically inadvisable the move may be (e.g., move the king forward) as long as it's valid by the rules of the game, delete duplicates (that is, count each move just once, not again and again in each branch that it's still possible to make it), how many moves are there?

Also, what is the most efficient way to tally up these moves? How far can we take this method, say, Black's second move, White's third, or further still?
Based on your comment, you are really counting the number of single moves possible in white's first two.  Each pawn has three possiblities, one space, two spaces, and (if it was advanced two on the first move) from rank 4 to rank 5, for a total 24.  Each knight has two for a first move and eight for a second move (including back where it started-is 1 Na3 2Nb1 different from 1Nc3 2Nb1?  Does that count as four moves?), total 20.  Each rook has two for a second move.  Each bishop has seven.  The queen has nine and the king three.  Overall total $24+20+2\cdot 2 + 2 \cdot 7 + 9+3=74$ possible moves in white's first two.  It depends also whether you think 2Bh6 is different from 2BxNh6 and 2Bxh6.  I have counted it only once.
I just worked out how many possible ways a chess game can progress through White's second move.  I got $8,902$, but I was outdoors and didn't use any paper so I may have made an error.

If we're just looking at White's second move and ignoring Black, I guess it would be... figuring backwards... it was $8,902$ and not $8,900$ because of the small number of moves Black can make that interfere with White, giving or subtracting a move.  So, $8,900$ divided by $20$ is $445$.  That's how many positions White can have after move $2$, but it's not clear whether that's exactly what you were asking.  The poster above may have answered what you meant to ask.