Vector graphics are excellent for logos.
Slides from class – a summary of graphics types
Using concept sketches to create ideas for logos is a really important first step.
Make sure you create your concept sketches first.
There are some guides you might fnd helpful:
How to use pathfinders to create a hut shape
Guides to get started:
More complex things:
The London Underground uses a design called a roundel as its logo. It's copied and used all over the world as it's such a cool design.
Can you make an animal face using vector graphic tools?.