Hi Liz,
I'm Jamee and I completely understand how unnerving it is to have your first pregnancy in Demnark. I have two boys now, one three years old and one three months old, so I'm fresh form the experience again and would be happy to help if I can answer any questions.
Before anything, tho - Congratulations!! After the initial doctor visit - my first visit was horrible too! - I actually had a great experience both times, so don't despair yet. It could just be the doctor you are at. I have a huge amount to say about pregancy care in DK, although I have no comparison to the US since both my kids were born here, but I'll try to stick to facts and you can ask for my opinion later if you want it. I actually live in Sønderborg, not Copenhagen, but I imagine the system is the same.
Firstly, if you don't like your doctor you can change and you absolutely should. If you have any friends in the area who have kids and really like their doctor, ask for the name of the clinic, then go to your local kommune and apply to change to that one. It will take 30 days, but it's worth it. I did that after hating my first dr. appointment and found an office I really liked. They even knew the clinic I was at in the first place and they seemed to have a bad reputation, so there are good ones and bad ones just like anywhere. There is good care to be had out there and you may just have a comparatively bad doctor at the moment.
Nextly, check your mail. Do you or your husband speak/read Dansk? If not, be checking for things from the local kommune and also from the midwife center and make sure you have someone translate them for you. There will be an appointment for your first midwife visit in the mail, and there is a phone number to call on the notice if you want to change the appointment. Call it and request an English speaking midwife. It makes all the difference if you aren't somewhat fluent in Danish.
Lastly, you are probably zoned for a particular hospital. Ask your midwife about a hospital tour. Usually the hospital has one for anyone pregnant at certain times each month and they will walk you all over the wards and show you where you will check in, the rooms where you will give birth and where you will go after the birth. Make sure you ask them to speak english or bring someone to translate for you. I did the tour in Sønderborg and felt a lot better about things after seeing a little more about how the actual day would happen and who would be present, etc.
That's all the major stuff I can think of initially. As you move into the second trimester and start going to your ultrasounds, the system kind of takes over with all the appointments, etc. and you should have plenty of people to talk to during your pregnancy. If you ever feel like talking to anyone else, though, I'm happy to chat any time.
Take care, and I hope your experience gets much better from here. Enjoy your pregnancy!
All the best,
Jamee Lyn Hansen
+45 4022 9582
PS Did anyone tell you about the nurse that comes to visit you after your baby is born? It's great - they keep an online record of size and weight for you to access any time and they are available 24/7 for any questions or concerns you have about your baby.