As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this MO question.
Here is a summary of the information in the previous question. For the second part of your question, the author (me) conjectures that for any finite set $S$ with all rational distances, no such point $P$ exists. As I noted in the comments, this is true when $|S|=3$, proven by Almering.
It is not known if there is a point with all rational distances to the unit square. However, it is known that there are no points at rational distance from all vertices of a regular $n$-gon, except perhaps when $n=4,6,8,12,24$.
Some more tangential remarks are that it is not known if there is a dense set of points in the plane with all distances rational, although it is conjectured that there is none.
Even more tangential, it is not known if every planar graph can be straight-line embedded in the plane with all edges having rational length, although it is conjectured to be true.