Well, the best two methods that most use are either shaving or using a hair removal cream. Though many say that they will burn the area, that's not true. If your skin isn't reactive to things such as this, you'll be fine. Shaving can cause some itching an irritation if not done properly, but it's nothing that you can't learn.
If you want to use hair removal cream, purchase a well known brand such as Nair or Neet, for their are many people that have used it without any bad consequences and they tend to have some for people with sensitive skin in case that's the kind you want to buy for your genitals. All you do is put some on the area you desire to become hair free and wait 5-10 minutes and rince it off in the shower or with a cloth. Read the back of the bottle for more information on how to properly use the product.
As for shaving, get a good brand of shaving cream you're acquainted with and make sure that there is enough on the area so that you don't nick yourself or have to keep going over the area. Also make sure the razor is sharp and new. Do like Hippie said and clip it short first so that there is not so much hair to go through. I'd advise shaving in the direction of the hair so there is not so much tugging, but it's up to you.
For more information, feel free to look online for methods on how to remove the hair and how to shave and use hair removal creams right. Learning more about the product helps too. Talk to people who have used it if you're truly concerned but reading online should give you enough information to do so. If you don't want to remove them, talk to your girlfriend and compromise just a low trimming. You don't have to do what she says for she should love you for you and not your body and dictate how your body looks. She can suggest what she thinks would look better, but that's all. It's up to you and what you decide is best for you and your body. Hope this helps.