Do:
Tell her you're happy to listen if she needs to vent. Sometimes, just talking to someone and getting things off your chest is helpful.
Say that you won't judge her for anything she has to say. She may not feel comfortable being super open with you at first since you're new friends, but she'll warm up to you over time.
Tell her to let you know if there's anything you can do to help her.
Clarify that she doesn't have to tell you anything if she doesn't feel comfortable doing so.
Promise her you won't tell anyone else what she tells you unless she says it's alright.
Ask about what she needs from you or feels directly even if it feels obvious or weird, since it's better to ask lots of questions than assume things. If she seems distant or touchy, she is probably feeling worse and could use some support.
Don't:
Try to relate to her situation if you don't have depression. This is essentially comparing depression to normal sadness and can seem like you don't understand how difficult depression is to deal with.
Get mad when she doesn't want to talk about something. It's not your fault, she just doesn't feel comfortable being so vulnerable with you yet and being upset about this would make her scared to tell you anything in the future.
Push her to tell you something if she doesn't want to. If she doesn't want to talk about something, you should respect that and change the subject.
Reveal sensitive information to other people. If she trusts you with information, you should respect that trust and not tell anyone unless she or someone else is in immediate danger.
Heavily focus on her depression. There's more to her than just her illnesses, and you should focus more of your energy on learning about the rest of her, like her hobbies and favourite things.
I hope neither you nor her ever need to use these, but here are two important hotlines:
Suiciide Prevention Hotline: 1-800-827-7571
Poison Control: 1-800-942-5969
If somebody is in immediate danger, call 911 instead.
Remember: She's still just another person! You don't need to be especially clingy with her, and it's not your sole job to ensure her safety. She has friends and family that will do the same as well, and you can inform one of them if something is wrong that you don't think you can help with.