Motorfiets 'motorcycle' has a cylinder capacity of more than 50cc, or is capable of speeds above 45 km/h. There are the driver licence classes of A1, A2, and A for motorcycles.
For two-wheelers:
For three-wheelers:
(Translated from this, downloaded from this page; I hope that I got it correct.) The rules are basically the same throughout European Union; this is what the European Union says about different classes of driver licence. Motorcycles have 'normal' registration plates like cars, but the dimensions are different. Helmet and insurance are compulsory.
Sorry I have no experience with motorcycles in the Netherlands. Here is an excellent website lazymotorbike.eu about motorcycling in the Netherlands and in some other countries. There is also a Dutch version of the same website: luiemotorfiets.nl.
Here are electronic maps of some motorcyling routes in the Netherlands from bestbikingroads.com, and routeyou.com.
This is a list of all(?) motorclubs in the Netherlands, and also links to other motorcycle-related websites: motorclubs.nl.
If you can go faster than 50 km/h, then you can go on an autoweg. If you can go faster than 60 km/h, then you can go on an autosnelweg. Motorcycles are not allowed on bicycle paths, and bicycle/moped paths. Motorcycles generally follow rules for cars; motorcycles have the same speed limits as private cars, and there are very few traffic signs that specifically mention motorcycles. The following is a traffic sign with a motorcycle icon: car and motorcycle icons in a white circle with red border (sign C12) – closed for all motor vehicles. Motor vehicles include private cars, motorcycles, and others like motorised handcarts, buses, and tractors. Not included are mopeds, (electric) bicycles, and handicap vehicles. (The bottom sign: uitgezonderd bestemmingsverkeer 'except "destination traffic"', i.e. you can only enter if you are visiting a specific address within, e.g. if you live there, visit specific people who live/work there, going to a specific place within for work purpose.)
Also see the pages for cycling, moped-riding, and driving for Dutch traffic rules in general.