@Macha Realistically...?
If you talked to HR they'd probably just log the complaint, tell her to stop, and you'd go on with a slight stigma of a complainer. No need to rush to HR first week on the job.
However, all HR processes also suggest telling the person that what theyre saying isnt work appropriate, and if you don't feel comfortable you can always talk to your boss. Which, right now, I'm assuming is your trainer.
If I were in your shoes and didn't want to risk being that person, I'd just ask the trainer for a quick chat, let her know that X person was making some pretty rude side commentary about homosexuality, and ask the trainer to just keep an eye out for it. Which IS a level of reporting it to HR, but puts it in the arena of where it matters most: mouthy person's immediate workspace.
At that point, it would be in the trainer's hands, you'd keep the rep of someone who would go to their management before blowing the roof off of HR, and any follow-up called out by your trainer would be between the employee, HR, and the trainer.