I am eager to try some some types of formatting, like in-line code, image customization, HTML support and a lot of other things to improve my posts. Does Stack Exchange have a sandbox page where I can try this without changing existing answers or questions?
Any new contribution that you start writing as a registered member can be considered a sandbox. As your edit it and it reaches a minimum required length, the system will auto-save it as a draft. Only you will see your own drafts as you return to that same page where it was saved. It will do that from then on automatically. I'm not sure exactly what triggers is, but it seems to be some combination of substantial change and time that has passed since the last saving of the draft. When it does, it will display a small notification in the left bottom corner under the input form but just above post preview:
Once draft is saved, unless you intentionally discard it with the link right of the post button, you can even navigate out of current page and return later to it and it will restore from the last saved draft. And post preview under the input form is WYSIWYG (What You See Is What You Get) and updates with any change as soon as they're made, so use the combination of drafts and previews for sandbox mode.
You can also use any post (question or answer) to try out different formatting by entering to edit it and then abandoning changes. Just don't also inadvertently submit those changes (as a suggested edit or a direct edit, depending on your site privileges), if you also don't want to create a new revision for it. Note that edits don't have this draft saving capability, that's just for new posts.
2$\begingroup$ Note that drafts only last for a week after last being saved. $\endgroup$ Dec 16, 2015 at 5:58
Feel free to use this answer as a sandbox, I've set it to CW, so anyone can edit it.
There's a formatting sandbox on the mother meta, and one for MathJax specifically on Math. (There are a few other sandboxes, no doubt, but not of any particular relevance.)