Ok, let me go through the steps myself once again, to make sure I have it exactly right.
(1) Only one page (page, not post) will show the slider, so you need to decide which page you want to show it. I will assume you want your home page to show the slider, since that makes the most sense. In your Dashboard, go to Pages/All Pages. This will give you a list of the Pages you currently have on your site. (If you do not have a Page, you must create one in order to use the slider.) Mouse over the Page you want to be your home page, and select the "Quick Edit" option. There will be some drop-down menus that show up: "Parent," "Template," and "Status." In the "Template" menu, select "Showcase Page." Selecting this as the template tells Oxygen that this is the page you want to show the slider. Click the "Update" button in the corner of "Quick Edit" to make sure your changes are saved. I suspect some of the problems you're having may be due to not clicking buttons for saving changes you've made.
(2) Now, in your Dashboard, go to Settings/Reading. The first item presented to you will be "Front page displays," for which you have two options: (i) Your latest posts, (ii) A static page. Select "A static page," and then in the drop-down labeled "Front page" choose "Home" (or whatever the title of your desired home page is). Scroll to the bottom and click "Save Changes."
So far, you have accomplished the following: In (1), you told Oxygen that you want to use the slider, and said which page you wanted that to happen on. In (2), you told your blog to land on that page specifically whenever someone types in or clicks on your URL. (I have not tried doing one without the other, so I don't know if you can have the slider without setting a static front page. I will assume you don't want that, though, since the whole point of a slider is to have an attractive main page of your website.)
(3) Now, you need to set Featured Images. Like I said, the images need to sized on your computer to be exactly 750x380. I think that Oxygen will crop them for you if they are larger, but if they're smaller nothing will show at all. Anyway, if you want complete control over what shows in your slider, make sure any pictures you want to show up are exactly 750 pixels horizontal and exactly 380 pixels vertical so that nothing gets cropped (or worse, doesn't show up at all). Once you have your images sized to these specifications, go to your Dashboard and select Posts/All Posts. On the list of Posts that comes up, mouse over a Post that you want to show up in the slider and select "Edit" (not "Quick Edit"). The "Edit" page has a box labeled "Publish." In this box, under "Visibility," select "Public" and check the box titled "Stick this post to the front page." (This is what makes a post "Sticky." By the way, this option is also available in the "Quick Edit" menu, where you can select the option for "Make this post sticky," but we also need to set a Featured Image, which requires the "Edit" menu, which is why I'm steering clear of the "Quick Edit" menu.) So far, you've told Oxygen that this Post should be one of the Posts featured in the slider. It is my understanding that unless you make the Post "Sticky" it will not show up in the slider. Now, you just need to tell Oxygen what image to use in the slider to showcase this Post. Still in the "Edit" menu, go to the box labeled "Featured Image," and click on the option "Set a featured image." Click the button labeled "Select files," and select the file on your computer that you wish to use as your banner image for this Post. When it uploads, scroll down to the bottom and select "Use as Featured Image." (None of the pre-set sizes available for you to choose will be 750x380, which is strange, but selecting the "Full Size" option results in the correct image showing in the slider even though "Full Size" purportedly re-sizes the image to 470x238 according to my menu.) Do not click the "Insert into Post" button, since right now you're not trying to insert the image into the Post; you're just trying to get it to show up in the slider. On the image window, click "Save all changes," which will bring you back to the "Edit" screen. Then, in the "Publish" box, click "Update" to save your changes.
Word for word, step by step, these are the exact measures I took to get my slider working. If that doesn't work for you, I suggest contacting DevPress, the people who made Oxygen, to see about troubleshooting.
Hope that works!