IF/VLOOKUP help

Hi, I'm looking for some help perhaps using the VLOOKUP function but I am not sure if another function would be more helpful.

To briefly explain I am trying to set up a spreadsheet which automatically calculates the start and end dates of project tasks, by looking at the order in which tasks need to be completed.

I have attached a spreadsheet to show what I am trying to do.

Thanks for you help,

Andrew