Take a look at the 2015 Social Media Image Sizes cheat sheet that contains the dimensions for various images across different networks.
Also read about creating your own gravatar and generate an image that will serve as your profile pic on different networks.
Think about your own brand and how you might portray it on these networks. Write a short statement (ie. no longer than a tweet) that encapsulates this brand.
Pick at least 2 networks that you wish to be on to engage with your target audience. For each create a banner and an avatar photo that reflects the brand you wish to portray.
Post the images to your blog, and if you wish, onto the networks themselves.