Getting her something romantic is a good way to go, but since this is your first gift, you need to show some restraint or you will overdo it and your gift can backfire.
Some ideas are a bottle of medium-priced perfume, her favorite type of fancy chocolate or candy, or a bouquet of flowers. Find out what kinds of these things she likes by taking a peek at what she already has, asking her friends or, if you are confident in your own tastes, simply picking what you think she would like. You should spend no less than $15 and no more than $50 on this kind of gift.
Make her first gift a fun activity, either for herself or the both of you.
A great idea is to give her a day at the spa, or even just a manicure on you. This has the added benefit of being unexpected for a guy to give, and very few girls will be unhappy to get such a gift. For a fun activity together, take her to see her favorite musician as a surprise, or if she is into sports, take her to see the team she likes. Other ideas could be hiring a sailboat for an afternoon, having lighthearted photos taken at the mall or taking a dancing lesson together. Be creative!
Food and Drink
Though you are probably already going out to eat together, a special meal or food can be an excellent first gift.
Find out what her favorite foods are and surprise her with a multicourse meal of those foods cooked by you in your apartment one evening. If she has a favorite wine or specialty food, you could find a unique version at a gourmet food store and present it to her wrapped in a classy ribbon. Another option would be a romantic picnic that you have set up somewhere quiet and pretty beforehand. If you put some thought and work into it, a simple meal can be elevated to the status of perfect first gift.
Most girlfriends love when they know that you have been paying special attention to them, and they like it even more when you show that to them in the form of a personalized gift.
This takes a bit more creativity than the other options, but can become a truly cherished item if done well. If there is a certain creative field you are strong in, use that to make a gift for her. For instance, if you can sketch, sketch her a quick but refined picture of herself, or if you're good with cameras, have a photo shoot with her as the star. You want to avoid getting too cliche or sappy this early in the relationship, so if you must write her a song or a poem, keep it light and fun.