Tragus piercing - would it be a good idea to use salt water if there's an infection?

Antibiotics & check: I am not sure what saline or salt water is going to do if you have an infection, other than help clean it out. I would recommend a topical antibiotic ointment like bactroban (mupirocin) to the area a few times a day and maybe start an oral antibiotic, if needed. The most important thing is to get this looked at by an ENT doctor who can prescribe some treatment options based on how infected it is.

Yes: Rinse out well with any kind of water. Throw away Bacitracin & neosporin: they are absolutely useless and are among the most common things people are allergic to. Frequently allergic reactions to these useless antibiotics are mistaken for infections because they are red & weepy.