Well first I'd say you shouldn't be embarrassed to exercise outside because I think more people should. Walking is great to do outside. Of course, baby steps and it may take a while for you to get to that step
As far as exercising indoors, exercise videos are a great option and designed for exercising indoors.
I have built a home gym over time and have various equipment including a swiss ball, a step, kettlebells, hand weights, weighted balls, weight bench (I started with a small one, I have a larger one now) and more expensive stuff including elliptical and weight machine.
Right now I'm following a program called "New Rules of Lifting for Women" which is an exercise program outlined in a book.