What are my rights if there is a ticket reporting error on the the travel agent's side? For example, what if I buy a ticket from airtickets that also includes baggage, but when I check-in at the airport, I am told that there never was any baggage included in the ticket. Is the airline obligated to take my baggage or would I have to pay extra at the airport?
This is really a law question regarding a breach of contract, and not a travel question.
What rights you have in this situation depends on the law where (a) the travel agent is domiciled, (b) your payment provider is domiciled [assuming you are not paying by cash], and (c) where you paid for the extra bags. Your question about "rights" is too broad to answer in this present form; but inevitably you cannot enforce them at the airport.
For practical travel advice, I suggest you pay the baggage fees on a credit card, and then dispute the payment later with your credit card company. Be sure to keep all the written documentation from your travel agent, as well as the baggage receipt from the airline, as evidence for the dispute. It is helpful if you paid for the baggage and the ticket on the same card. Most card networks will refund monies of this size to the consumer without argument.
As an aside, your baggage allowance will be encoded on the e-ticket (or printed in the "ALLOW" column on the far right of your paper ticket). It is unusual for a travel agent to get the allowance wrong because there is no guessing, it is stored in one authoritative place.