How to change the size of nodes?

The node size normally depends on the size of the content. Your node is empty so it is relative small. If you want to set the size manually use the minimum width and/or minimum height or set both at the same time using minimum size.

The circle shapes just uses the greatest of both values, otherwise you may be interested in the ellipse shape.

\begin{tikzpicture}
     \draw node[fill,circle,minimum size=2cm] {};
\end{tikzpicture}

If you want to mark the node even smaller than it is in your question, set the inner sep value lower. It causes an empty node to still have some size. Best set it to 0pt and then use minimum size.

If you only fill the node and do not draw its border (as you do in your example despite the \draw command (that does not affect nodes on the path)), you might also set the outer sep to 0pt, so that paths will actually connect to it.
If you do not reference the node later, you could actually just use the circle path operator, where you can specify a radius very simple: \draw (0,0) circle [radius=<value>];

\begin{tikzpicture}
     \draw  node[fill,circle,inner sep=0pt,minimum size=1pt] {};
\end{tikzpicture}

\begin{tikzpicture}
     \draw  node[fill,circle,scale=0.3]{} (0,0);%
\end{tikzpicture}