How can I tell if a rim is bent or if the wheel is just out of true?
The common qualification here is an important one: truing is a skill that benefits from practice and experience (chicken, meet egg). Your wheels are important, and it's important to have somebody who knows what they're doing at least check your work if you're just learning (or if you're disinclined to trust your well-being in dense traffic to wheels you trued yourself).

Having said that, a rim that isn't bent should have more or less even spoke tension across all the spokes when it is true. On the other hand, a bent rim will require higher tension in some spots and lower tension in others. So, noticeably uneven spoke tension in a true wheel is one sign that the rim might be bent.