One of the best ways to save money when installing a bay window is to upgrade to highly energy-efficient glass and frame materials. Choosing fiberglass window frames and double-pane or triple-pane glass will drive up your up-front costs, but they can also save you quite a lot on energy bills over cheaper, less insulative materials.
You can also save money in the long run by choosing a reputable window installation company that provides a long-lasting warranty. Higher quality companies will tend to be a bit more costly, but again, they stand to save you money if you run into issues with your window in the future.