Belgium is a relatively small country. One can drive out of it in one hour from almost anywhere, and not even half an hour from most cities. The train is also a good way to get out of the country for the day. Lille, for instance, is only 35 minutes away from Brussels by TGV.
Although located in other countries today, Lille, Maastricht and Luxembourg were historically and culturally part of Belgium (Spanish/Austrian Netherlands). Lille is the capital of French Flanders and was once an integral part of the County of Flanders until its annexation to France by Louis XIV. French people living between Lille and Dunkirk have Flemish surnames and some can still speak Flemish despite 350 years of Francisation. Culturally the whole of Nord-Pas-de-Calais is more Belgian than French.
Maastricht and Luxembourg were part of Belgium when Belgium gained its independence in 1830 (=> see Belgian history). They were ceded back to the Netherlands in 1839 in exchange for the recognition of the independence of Belgium. Luxembourg got a separate independence in 1890. Aachen, though never Belgian, shares an intimate history with nearby Liège, cradle of the Carolingian dynasty. Liège (Lüttich) remained German until 1792.