Get the job offer in writing. I always ask for the job offer to be written so that it is less difficult for it to be rescinded for whatever reason.

As for past mistakes at your other job, call the HR department of your former employer to find out what they will disclose. Most places have a policy of only disclosing your job title and dates of employment, and in some cases, maybe if you would be eligible for rehire. If they do that, then this should put your fears to rest. If you were not there long, then leave the job off your resume and job history (unless you are being asked for ALL employment history, this happens with things like government jobs that require a security clearance, etc.).

You will be fine :)