Should I attempt to get to a potentially damaged trap and check it out myself?
I question whether busting up the concrete right away is the correct thing to do, it seems a bit premature to me.  I would try to get a camera with a locator down there so you can at least get an idea of where the sewer goes and then break up a localized section of concrete if the image is too poor.

Most of what is required to do this is just manual labor - busting up the concrete and hauling it away, and then digging until you get to the sewer - could be a few inches or several feet deep and your concrete is probably about 4" deep.

You would most likely want to rent a jackhammer from your local tool rental.

Assuming it was a crushed pipe, you'd need to remove the old section and replace it.  You don't mention what the sewer is made of, but this might involve cutting and gluing PVC, or be more complicated and involve connecting different types of materials like clay or cast iron to PVC (as the patch).  Each one of these would be a good candidate for a new question.

You'll need to be mindful of any other buried utilities and if it is clay, expect to find some tree roots (sometimes really big ones!).

Definitely DIY-able but be prepared for a lot of back breaking work, and assuming the backup doesn't drain on its own (usually some liquid can still move), a lot of filthy sewage water.  Make sure you take precautions to protect yourself - gloves, eye protection, mask, rubber boots.