You should not use an ellipsis like this in formal writing, as backed up by Grammarly. (The exception, of course, is when indicating a portion of a direct quote has been removed.) You don't need to include any special punctuation here; "such as" already indicates that you are only planning to give a list of examples, so you can just write "such as X, Y, and Z." And "for example" is a good synonym if you want to spice up your word choice a little. Formal writing tends to discourage the other alternatives, such as ending the list with "etc" or "and so on" since they are a little informal. (But again, including either of those would be redundant if you're also starting the list with "such as" or "for example".)
There are few styles guides dedicated to other types of writing (e.g. novel writing, where there are ellipses aplenty), and I couldn't find anything that addressed this in particular. But I think the rules for formal writing work just as well in that context too, specifically this one:
Omit any punctuation on either side of the ellipsis, unless the punctuation is necessary to make the shortened quotation grammatically correct. — The Punctuation Guide
This would mean that you would omit the comma:
I still have to do so many chores, such as cleaning the toilet, taking out the garbage, doing the dishes...