Yes, holders of an EU Blue Card are permitted to travel to other Schengen countries within the EU for tourism purposes without a visa. For up to 90 days within a 180-day period, you may even conduct business in other EU countries with your Blue Card, provided that this activity is related to the job for which you received the Blue Card. This allows you to participate in business meetings, negotiations, or training courses in other EU countries, among other things. If the EU member state to which you wish to travel for this purpose is not part of the Schengen area, you may need to provide additional evidence in order to be allowed to enter. We therefore recommend that you always check with the relevant authorities in the destination country to find out about the rules that apply there.
After you have stayed in Germany for one year with your EU Blue Card, you can leave the country at any time to obtain a Blue Card and work in another EU member state. After entering this other EU country, you must submit an application for a Blue Card to the immigration authorities there as soon as possible, at the latest within one month. In order to obtain this, you must meet the requirements of this country for the issue of an EU Blue Card, but this should not usually be a problem as you have already met the requirements for the issue of a Blue Card in Germany.