By signing your name to that credit card or car loan you guaranteed payment to the bank. That means you promised the bank to pay back the loan if the primary borrower defaulted. Both parties who signed the loan are 100% responsible for the entire debt- not a 50/50 division. That default usually has no conditions. That means if the primary borrower died, skipped town, lost his job, or has money but is choosing not to pay the creditor can sue you. As co-signor you are just as responsible for the debt as the primary borrower without the benefit of the goods purchased.
So what can you do to protect yourself? Well- if possible contact the creditor and see what can be done to salvage your credit. If there is a car, you may be able to arrange a voluntary repossession to help pay down the loan. If there is only credit card debt you may be able to arrange a payment schedule. Of course you can always sue the person you co-signed in an indemnification lawsuit, but often they don’t have any money. This is the reason the creditor required a co-signor in the first place. The bank practically expected a default and wanted a backup person to sue.
So can you sue the person who was supposed to be making the payments? Sure you can, but they likely don't have the money or else they would be making the payments. If the person you co-signed for files bankruptcy you should call a bankruptcy attorney to find out what this means for you. If they filed Chapter 13 then the co-debtor stay should protect you from getting sued during the active bankruptcy. Any balance due at the end of the plan payments would be your responsibility. If your co-debtor files Chapter 7 on an unsecured debt then you are likely left holding the bag with no protection. If the debt is secured debt and they intend to remain current or reaffirm the debt then nothing may change. As mentioned earlier if bankruptcy arises- best to speak with a bankruptcy attorney.