I am in no way a social media genius. However, I am sharing here common images sizes I have gleamed from around the web, having tested them myself. I also recommend building these items yourself, using a program such as the Free & Open Source Inkscape, especially if you are interested in creating in Open Source Software.
Header Image (1500x500, 1263x421)
Many sites list the header image to be 1500 pixels wide by 500 pixels high, however they also mention that you can make this 1263 pixels wide by 421 pixels high if you wish to not have your header image resized. This second set of dimensions is what I personally use when creating Twitter header images for my accounts.
Avatar (500x500 square)
While I say square, keep in mind that most social platforms including Twitter have switched to showing your avatar as a circle. So keep that in mind when creating an avatar, especially when they include text.