You may want to broaden your search to include some other towns in the area. Heidelberg is expensive especially near the town centre. Neighborhoods within or near HD that tend to be more expensive are Handschuheim and Neunheim. However, places like Ziegelhausen/Schlierbach, Kirchheim, Rohrbach, and Dossenheim are further from the town centre and tend to be cheaper (though you'll still probably be limited to an apartment). Pfaffengrund and Eppelheim are also decent areas that are relatively close to the International School. If you can't find something in any of these places that meets your needs, you could always look someplace slightly further out like Leimen or Nussloch. These communities are generally cheaper but may be too far out for you. We certainly found this to be the case, but we also don't have a car. It really depends on what you're looking for (sq. m, how many rooms, etc) and if you're planning on having a car. Here are a few threads that have addressed the neighborhoods or house hunting in Heidelberg.
http://www.toytowngermany.com/forum/index.php?showtopic=144202
http://www.toytowngermany.com/forum/index.php?showtopic=94198
http://www.toytowngermany.com/forum/index.php?showtopic=130016&st=0&p=1606928&fromsearch=1&#entry1606928
Die Business Nanny made a good point about rent. You'll often be given prices in Kalt rent (without heat) + Nebenkosten. Apt/house prices with heat are called warm. You'll also have to ask what is included in the Nebenkosten when you look at places. If you haven't already found it, there's a nice
TT wiki thread on the subject of apartment rental, and
this is a link with definitions of many housing ad acronyms.
Good luck