there are different questions in your request, so I will try to divide them.
1) What can you do in your 1 month waiting period ("karensperiode") as graduate?
In the 1 month waiting period (karens) you can do whatever you like to do. So you can go to another country on holiday, visit family or what else you like.
Graduates with a waiting period are not required to register at the Jobcenter (Jobnet.dk) before the month has passed. If they do this anyway, they will be called in for talks with the A-kasse and the Jobcenter.
The "waiting period" begins to count from 19. december 2019 and will be gone on 19th January 2020.
From that day 19th January 2020, you can get dagpenge. However it requires that you are in Denmark, and are available for the labor market. i.e seeking job and go to meetings in Jobcenter/A-kasse..
If you want to have more holiday after the waititng period, until 31st of January, you can do that but must notify the Jobcenter/A-kasse about your holiday at latest 14 days before.
So in your case you should be able to receive dagpenge from the 1st of February.
I would anyway recommend you to get this confirmed by your own A-kasse.
2) Can you go away for the weekends from Denmark?
You can receive dagpenge during a weekend stay abroad if you travel abroad at the earliest after normal working hours on Friday and returns to Denmark in good time so that you can familiarize yourself with your post and in time be able to start a job or offer, to which you may may be referred, at the commencement of working hours on Monday. It is a condition that you was able and willing to stay in Denmark over the weekend and take over work if you had been referred to a job.
Hopes this clarify things a little for you. Otherwise you are welcome to post comments here.
Anders Weber, www.a-kasser.dk