Usually I have seen users having trouble to download and install trial edition of Project 2013. Following steps will help you to download and install it.
How to download Microsoft Project 2013 trial
1) Open the link http://technet.microsoft.com/en-in/evalcenter/hh973401.aspx and click on “Get Started Now”
2) It will prompt you to enter your windows live ID and will present you the page with the “Personal Information”. On this page you need select whether you want to download 32 bit or 64 bit version of Project Professional. Some confusion can arise if there is a mismatch between a version of Office already installed on your computer and the new Project Professional version. First, check whether you are running 32-bit or 64-bit Office. Then, when beginning the Project download, choose the corresponding version of Project. Refer to screenshot below to select the correct version
3) On the next page, you will get the product key which you need to use during the installation. Make a note of the product key and click on the Download load button after making the language selection.
4) As next step you need to specify the download location to save the ProjectProfessional_x86_en-us.img and click on Save.
5) The Download will start and you will see the progress in IE window
6) After downloading ProjectProfessional_x86_en-us.img successfully, you need to use one of the below steps to install the Project Professional
a. Mount the image as an ISO virtual drive. There are popular tools such as Virtual Magic Disk to mount ISO/IMG disk image file as virtual CD-ROM drive and then install from the ISO file.
b. Extract .img. You can extract the .img into a directory using any tools that unzip the .img format (e.g., 7-zip) and then install from it.
Hope this helps!