To have an image show in the excerpt you need to set it as a "Featured Image".
Go to the top of you post page in Edit mode and look at your Screen Options. Make sure "Featured Image" is checked so it shows up on your post page. From there you can set which image you want to have show up in the excerpts.
You will need to go back to all of your posts and set the featured image for each one. Or, there may be a plugin that will do that for you. From here on out, when you are writing a new post you can set the featured image when you are placing the image in the post itself. Look down at the bottom of the Image screen. Right next to "Insert Image" there will additionally be an option to Set as Featured Image. If you forget to click that at that point then you will have to use the "Set Featured Image on your post edit page.
Ps. Axx, it is always a good idea to include a link to your site in any help request. It wasn't necessary for this answer but often it is a big help. It is just a good habit to get into.