It's a good start. Now comes reality.
The moves you missed are castling, en passant, and promoting a pawn. Also, you can't move a piece out of the way that is pinned to your king (in some cases, you can move it as long as it continues to block the check). If the king is already in check, the only legal moves are ones that removes the check (block, capture the checking piece, run away). If double check, then both checks must be resolved (which means the king has to run away, possibly capturing one of the checking pieces, if possible).
I also agree with JosephP's idea of a MoveType class, although your offsets arrays may be enough (with an added flag for one time move, move until blocked, move if capture, move if no capture, move if en passant, etc.). Let the base Piece class do most of the moves and let the individual pieces just tell the base class what directions the piece can move. I'd let the board do the validity testing for the king being in danger, as well as remembering what square (if any) en passant is possible for. Castling involves checking various squares for opponents checks and emptyness, as well as the rook and king remembering if they have moved before (which can also be used for a pawn moving two squares).