Can I get in trouble for doing jobs under the table, even though I'm underage?
"Teenage taxation" is a complicated area. Depending on the kind of work you're doing, you could be considered either a household employee (in which case you should receive W-2 income) or an independent contractor, in which case you are obligated to pay self-employment taxes.
It all depends on a number of factors, including the amount you earn, the work you are doing to earn the money and the extent to which you are advertising your services.
To start your own business, you'll need the permission of a parent or guardian, who will (at least nominally) need to be in charge of the business, as teenagers are not, by law, able to form businesses or enter into contracts. Speak first with your family accountant about your tax situation and to get guidance on whether a separate business entity is really needed from a tax and financial perspective.