What should I do at the small chessboard?
You need another chessboard of sorts before you can do anything here.
Look in the desk at the far end of the hallway, near the stairs down, to find a piece of paper with a chessboard pattern on it.
Put the paper in the chessboard, and you'll now be able to press the labels at the bottom of the board.
Perhaps this is somewhat obvious, but you need to press each of the labels A-H in the appropriate order.
Those labels correspond to the positions of the pieces shown on the left of the board.
Dracula probably wrote down a combination as long as this one.
Look in his notebook and use the vampire glasses while looking at the chessboard page to see the positions of the pieces.
Just enter the column letter of each of the pieces from top to bottom.
This gives you the combination F, B, H, E, A, C, G, D.
When you've entered this correctly, you can remove the paper you put inside the board and find some holes punched. These will be important later.

How do I get across the chessboard at the bottom of the stairs?
There's only one safe path across the board.
It might help if you found a map.
Or made one, to be more precise.
You can do so by getting past the vampire lock upstairs and then using the chessboard that's hidden behind the lock.
When you do everything correctly at the small chessboard, you can reclaim your chessboard paper, only with holes punched in it now.
The holes indicate the only safe squares to stand on.
It would probably help you to know that the colors of the squares are all reversed -- that is, what's a black square on the paper is a white square on the floor and vice versa.
The stairs take you to the bottom of the board, and you're trying to get to the top on the other side. Just follow the path as indicated on the paper until Dracula finds you.