It is pretty simple to repair your broken iPhone screen, there are multiple instructions about it on the web. Just google your specific type of phone. Inspired by my friend gaiageek I used these instructions and it worked really well:
http://www.wikihow.com/Fix-an-iPhone-3G-Screen
I admit, it was a bit nerve wrecking and I had my repair-it-all dad with me who is more specialised in cars and was amazed himself that we could repair a smartphone. But with time, humour and the right size screwdriver we made it. It is also a good idea to consider replacing the battery as this is only a few more steps and can increase your phones value.
There are multiple repair packets, I bought my iphone screen for around 8 EUR and the battery for 10 EUR on ebay.com and ebay.de. If the LCD part is broken it is slightly more expensive.
Reduce.Reuse.Recycle.