Is it required to contact approach when flying VFR into a Class D airfield?
No it is not required to contact approach. Unless you have to go through a larger D, C or B airspace to get there. I don't know of any larger Ds in the US, but there are plenty of D-airfields underneath C and B airspace.

In Germany (and I think around Europe, not sure though), there's a distinction between the tower-D and the approach-D (the former being called D-CTR, and the other just D), to enter the approach-D you'd have to talk to approach, but if you can sneak under/around it, go ahead and call the tower directly, let them know where you are, and they'll tell you how to enter their airspace. In the US, they'll probably just tell you to enter the pattern (like "N12345, enter and report the right base for runway 5").

In Europe you need an explicit clearance and they'll usually tell you how to enter (like "DEFGH, enter control zone via Echo, report Echo", Echo being a reporting point on the chart, Echo usually being to the east, with November, Whiskey and Sierra being north, west and south respectively (usually, that is, there may be more, and they may be placed differently, but they're always on the sectional or the terminal area chart)