How do I know if it is broken?

If the lens is properly attached to the camera, definitely has autofocus turned on, and still won't focus, that's a good indication that the lens is broken. But you're a long way from that conclusion.

Or am I attaching it wrong?

If you're new to using the camera, there's a strong chance that you're not completely connecting the lens. The lens has a set of electrical contacts that only connect to the camera body when the lens is fully mounted. If the lens doesn't do anything at all, it's likely that those contacts aren't engaged.

As @ousullic said in a comment, the lens should click into place. There should be a red dot (or possibly a white square, depending on the lens) on the back of the lens. You want to line that up with the corresponding mark on the camera body, seat the lens on the lens mount (the metal ring on the body), and gently twist until you hear a soft click. Once the lens is properly attached, you won't be able to turn the lens body counterclockwise to remove it without first pressing the lens release button.

The really important thing to know is that the mounting hardware on both the camera and lens is made very precisely, and they should go together smoothly; you shouldn't need to force anything on the camera or lens into position, and that goes double for mounting the lens. If the lens doesn't feel like it's sliding into place, stop and re-evaluate.

This is my first slr camera and I am totally thinking I bit off more than I can chew if I broke the lens the first time I try to take a picture.

Give yourself a break -- you're new at this. Unless you dropped it, it's unlikely that you broke the lens. In the unlikely event that the lens is broken, it can likely be repaired.

If you do suspect damage, get some help. A friend or neighbor who has some experience with photography should have no problem attaching the lens and checking for damage. Camera stores are increasingly difficult to find, but if there's one near you they can surely help as well. (Remember to give them some business when you can so that they'll be there next time you need help!) As a last resort, you can also send your camera and lens to Canon to have it checked. Contact them first so that you know up front if they'll charge you.

If you're feeling overwhelmed, consider taking a photography class to get the basics down and build your confidence. If you can string together enough words to write an articulate question, you can learn to operate a Canon T5i, but it often helps to have someone show you the ropes.