For web, go with PNG for most images. For backgrounds, however, use a JPEG with no more than 80% compression. And, basically, avoid GIFs unless you need to update your Tumblr page.
Of course, image formatting for printing can get a bit more complicated. However, when it comes to card printing, High-res JPEGs and PNGs work fine—so long as you make sure to save everything at the highest quality as possible. Space is no concern. If you have any more questions, don’t hesitate to contact us!