I am looking for software that will let me draw a very simple design and then convert it into different sizes and rotations without losing resolution. The goal is to lay a number of these out into a pattern. I'm not sure if I need to go for something as sophisticated as Illustrator. Will you kindly make some suggestions please?
I would suggest you to use Inkscape, since it is easy to use and free (Open Source), therefore easy to try. It's a Vector-Editor, so scaling is no problem.
What you need is any kind of software that handle vector graphics. Here's a long list, pick one that meets your criterias (price etc.):