When dealing with breakups, most people can't help but have the feeling of "I want my ex girlfriend back". They start to think and wish that they could have done things differently to change the outcome. They might even write letters to apologize or profess their love in a desperate bid to salvage their relationship.
Doing all these aren't going to help you at all because you can never really know the real reasons behind the breakup. Women by nature are more emotionally driven and may not even know why they are acting or feeling the way they do. It is better for you not to lose your energy trying hard to figure out what went wrong.
The first answer to the question "what should I do to get my ex girlfriend back" is to let go of the relationship. Forget about it for a while. Channel and divert your energy into some other things. Forget about love in general, go out have fun, meet new people, pick up a new hobby. Make new changes to your life, it may take a couple of weeks or months. It will work wonders for you if you take some time away from your ex.
The effects of forgetting your ex for a while and make new changes to your life can only benefit you. Your perception on your relationship may be different now and you will have a clearer picture on what you want to do with your ex. Do you want to put her behind you and move on? Or do you still want her back? If you still want her back, it's time to find out how she is feeling about you.
Now that you have spent enough time away from your ex, she is probably missing you the way you're missing her. It is your chance now to take the initiative to get your girlfriend back. You can ask her out for coffee but please don't simply beg her to come back to you. You need to do it a subtle and non invasive way. Play a little hard to get but don't overdo it. Show her that you are doing fine without her. This will inspire her to want you back.
It will become apparent now if you're meant to get back together with your ex. Take things slow and easy. Don't over read things too much as it may blind you from acting right to get your ex back. Play them cool and you will have her in your arms again.