Creating effects/transitions in AviSynth or Lightworks
You will need so-called video composition software such as Adobe After Affects or even Blender.

A few years ago Combustion was available which was very good for things like this, but Autodesk came along and purchased it, sucked it dry of blood, and put it in a drawer never to be seen again.

AE comes with a price tag, while Blender is free (site http://www.blender.org appear to be down when I tested it now - see Wiki link above for information on it). Both have a pretty steep learning curve though if you never used composition software before. But there exist a lot of free learning material on the web to help you up and going.

Update: Adding some links to resources that hopefully can be of use:

After effects, layers, compositions - This will show how you can first create elements, then how to place them with different position and angle onto another video. The more relevant part starts from about 8:30 or so:
http://library.creativecow.net/articles/rabinowitz_aharon/Jumbotron/video-tutorial

After Effects introduction:

This site gives you bucket loads of video tutorials on AE and composition effects, layers and so forth:  
http://library.creativecow.net/

Video editing with Blender (part 1):
https://www.youtube.com/watch?v=1sODml0PBlo

Special video effects with Blender:
https://www.youtube.com/watch?v=x3ytZrGJ1ZQ

Hope these will be of help!