My prototyping experience is basically limited to solderless breadboard. Did do some wirewrapping years ago where the pins needed to be inserted in the holes.
I've never used the prototyping areas I've seen in photos of boards like this... do you have to push wires and pins thru and solder them on the back?
Yes, or solder in sockets or bend the legs of components and push them in. Wiring can be either the component leads themselves or wires top or bottom.
Look at the photos for this project. Note it might look messy but it is good wiring.
http://www.thebox.myzen.co.uk/Hardware/Mini_Monome.html
There are two basic classes of prototyping board.
Matrix Board
These boards have a grid of holes, each one with its own metal ring (known as an annular). To use them you can either put multiple leads and pins in each hole and solder the back (if there is room), or in adjacent holes and bend the leads over at the back so they meet, then solder them all together into one blob.
Bussed Boards
These have the same holes as the Matrix Board, but the holes are linked together in certain combinations. Some have a couple of strips down the centre to use as power, then groups of three holes linked together. Some have the holes tied together in pairs. There are many combinations. These are tidier and easier to work with as you don't have to do as much bending over and bridging of pins and leads on the back, but the component density will be slightly lower.
Some boards also contain surface mount component areas (see my eBay shop for one such example), which can be invaluable given todays push towards components only being available in surface mount formats.
Thanks
I find it convenient to solder all the components, maybe bending some leads over where convenient. But for the most part, solder and clip short. Then use 30AWG wire wrapping wire to make connections between solder pads. Like so:
Mmmmm... Spaghetti... 
An example of stripboard.
