How to wire up a 3-wire load cell/strain gauge and an amplifier?
The strain gauge is a variable resistor, so your first idea may to build a resistor divider with a second, fixed resistor to detect the variations as a change in voltage.
Unfortunately strain gauges are very insensitive variable resistors, whose resistance changes only very little when weight is applied to them. A resistor divider is not nearly sensitive enough to detect the changes. So we need another approach.
A Wheatstone bridge is the solution.  

  

The strain gauge together with R2 still form a resistor divider, so how is this different? Let's assume that all resistors have the same value, equal to the strain gauge's resistance at rest. Then the voltage across the voltage meter will be zero instead of half the power supply. Since our reading is zero referenced we can amplify it easily to get a higher sensitivity for the complete circuit.
Oli mentioned the differential amplifier, but this isn't going to be enough. We don't want to affect the reading by putting a load to it, like the differential amplifier would. We need an instrumentation amplifier, which is a differential amplifier with a very high input impedance. This is the most used instrumentation amplifier configuration, 



which uses a single resistor (\$R_G\$) to set the amplification. You'll have to set the amplification to a high value, possibly somewhere between \$\times\$100 and \$\times\$1000 (not very clear; the so-called strain gauge datasheet simply stinks).  

Now, how do we connect the strain gauge, because it has three wires, not the two like in the above schematic? Again the datasheet is no use here, but you probably connect it like this:  



This way of connecting compensates for the wire resistance \$R_{WIRE 1}\$, which otherwise would affect the reading.  

Another possibility is that the wires represent the top, right and bottom point of the Wheatstone bridge, resp. Which one it is can be easily determined by measuring the resistance between wires. In the first case you'll have no resistance between \$WIRE 1\$ and \$WIRE 3\$. In the second case you'll measure an equal resistance between red-white and white-black (you may have to switch wires. Again, the datasheet doesn't help).

You connect the voltage meter connections of the Wheatstone bridge to the inputs of the instrumentation amplifier.



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