Easter is a time where families visit their extended families, where you take the kids outside, or for short trips. Historically it is one of the highest bank holidays and protected by special laws.
Shops
In general, places that sell stuff will be closed on Friday, Sunday and Monday for the Easter weekend. That includes supermarkets, department stores, clothing stores and so on. On Saturday, they will be open because that day is not a bank holiday.
North Rhine-Westphalia does not allow an open Sunday on Easter Sunday.
The exception is train stations and other tourist and travel hotspots. Cities can have regulations that allow shops in those areas to be open, but that does not mean they will open. Some stores in Duesseldorf Hauptbahnhof open on Sundays from 6am to 10pm. Neither that website nor the website of the chain I linked to says whether they are open on bank holidays. This answer says that a similar store in a smaller Berlin train station was open from 1pm to 6pm on Christmas, which might be an indication that some stores also might open during Easter in stations.
Gas stations will be open. They usually have a small shop attached that offers bottled drinks, sweets and snacks, newspapers and cigarettes. Sometimes they also offer a bit more grocery-type things. Some also have a bakery attached. Items that do not have a regulated price are very expensive there.
Bakeries could be open, but not all of them will be. They might handle it like Sundays, so they might open later than usual (i.e. at 9am) and close at noon.
Again specific for North Rhine-Westphalia, bakeries are only allowed to open on the first of two consecutive bank holidays. That means they can be open on Easter Sunday, but not on Easter Monday. Bakeries in train stations are exempt from this regulation, just like shops, because they sell travel provisions. Other states may also have varying rules for this.
Restaurants
A lot of restaurants will offer lunch only on Friday, Sunday and Monday. They might be closed in the evening on these days. Bars might be open in the evening, but I would not count on it as Easter is the type of bank holiday that has the highest percentage of additional pay of all bank holidays in the hotel and restaurant field in Germany.
Fastfood chains like McDonald's will likely be open.
On Saturday, most restaurants should be open according to their regular hours as that's a normal Saturday.
Partying
Discotheques and clubs will most likely be closed on Friday and maybe on other days because of the Tanzverbot. That's an old law that forbids dancing on certain religious holidays. It does not exist in all states. Most Germans do not know even that it exists, but it's very real.
In North Rhine-Westphalia, dancing is not allowed on Friday (all 24 hours) and on Saturday from midnight to 6am.
However, it's very typical in Germany to have Easter fires on Saturday evening. Villages do that, but a lot of cities also have it in individual districts. Imagine a huge bonfire with beer booths and conversation with locals. In my opinion, that is worth seeing. Here's an online newspaper article about Osterfeuer in Duesseldorf in 2016. But keep in mind that places with a lot of alcohol attract stupid behavior. Apply common sense, and avoid drunken people. Don't get drunk yourself if you're not with a group of local friends.
Sightseeing
Touristy things in very touristy areas might be open. Looking at churches like the Kölner Dom however might be more restricted than usual because they need those for actual worship at times that usually do not have services during the Easter weekend.
Parks and recreational areas that have opening hours should be open.
I do not know about museums. Those will be city-specific.
Travelling
Local public transport (buses, trams and so on) will mostly work on the weekend schedule. Deutsche Bahn has special schedules that should be available a few month in advance online, just like any other regular ticket.
2017 cannot be booked yet at bahn.de, but I checked Easter Monday 2016 and there were ICE connections from Duesseldorf Hbf to Hamburg Hbf every hour starting from 14:52 on 27th March 2016. There was also a regular IC connection, but it seems there were no earlier trains.
From experience I can say that traveling with Deutsche Bahn on bank holidays, especially the family ones like Christmas and Easter, is interesting. If you like standing in a train where it's not possible to fall over that is. A lot of people will travel at the beginning and end of the long weekend. If you need to travel on Monday, reserve seats early. Remember that train tickets in Germany do not usually include reservations or the right to a seat (1st class excepted).
The Autobahns might be full. If you travel by car, plan for dense traffic at the beginning and end of the long weekend.
The first workday after the long weekend (that's Tuesday) however should be pretty regular in terms of traffic and fullness of trains. Most people will go back on Monday evening.
