Going to assume heterosexual couple and the grad student is female.
She has some sort of money because she is paying rent, utilities, etc currently even if she has a roommate.
One option is to split rent and utilities 50/50. I have heard of couples doing it proportionally so if you make 10x more than her, you would pay more along those lines. So if rent is $1K then maybe you pay 900 and she pays 100?
Chores around the house is a big one. It just needs to be something you both consider to be fair. You could each do your own laundry for instance. When I moved in with a woman, I told her how much I hated cleaning toilets, but doing dishes and the floors and windows was okay. So we split up the chores. I cut the grass (something I actually like to do). We even ended up combining laundry although she had a separate bin for things she was afraid I would mess up.
Cooking? We split it somewhat with the other person doing clean up. But we also cooked together and then cleaned up together or one of us would take it on and let the other chill.
Time apart. You now have the luxury of when you want to be alone to go to your place. When you live together you need to be clear with space. My ex and I are both somewhat introverted so we were okay being in separate rooms. Once we were watching the same show on tv in different rooms but just needed to be alone. We realized this was important and necessary for us, but it seemed strange to a lot of friends.
Having friends over. You each likely have a friend or two that the other person does not like. Now you can just hang out with your buddy and your place. Well, if your GF does not like him then do you go over to his place? Part of that need to not become dependent on one person for everything.
Also, your friends might start to think you cannot do anything without your partner now that you are living together.
What is going to be the level of agreed clutter or messiness or dirt that is okay? How long can dirty dishes stay in the sink?