Adding vertical space at the start of a page

The space added by \vspace is deleted at the beginning of the page, as you have seen. The command \vspace* adds the space that is not deleted.

\documentclass{article}
\begin{document}

\vspace*{250px}
I want to be further down on the page :(

\vspace{250px}
Haha! Sucker :D
\end{document}

Spaces at page breaks are swallowed, as usually one doesn't want them. TeX starts counting from zero, so the first page is after a page break. :)

The solution is to issue \vspace* instead of \vspace.

I recommend not to use px which is not what one with a CSS background would expect. Use rather cm, mm or in or multiples of \baselineskip:

\vspace*{2cm}
\vspace*{4\baselineskip}

The default value of 1px is just 1bp (where 72bp = 1in); it's a special unit of measure that can be tailored for specific applications concerning on-screen only documents. For example, to make a document as wide as a 1200 pixel screen at 96dpi, one can pass geometry a paper width of 1200px by

\pdfpxdimen=1in % just to start the computation
\divide\pdfpxdimen by 96 % 96 px are now 1in
\geometry{paperwidth=1200px}

For "paper" document, this is irrelevant.

In addition to the flurry of \vspace* suggestions, you could also issue a "nothing" command to allow subsequent \vspaces to typeset as expected. This "nothing" command could be an \mbox{} (an empty box) or \null, provided that you leave an empty line (or \par) to be in vertical mode.

More specifically, you would need

\documentclass{article}
\begin{document}
\mbox{}% or \null; note the blank line below. Alternative, add \par on this line.

\vspace{250px}
I want to be further down on the page :(
\end{document}

although \null is a better choice - it does not require the blank line or even \par.