Basically these sensors have a resistance that decreases as pressure is applied to their contact area. From what I can tell the decrease in resistance is a roughly hyperbolic (1/x) function of the force applied, and the change is basically from 5 MegOhms down to about 300 kOhms ("pressing hard" from the SF page).
If you use a 200 ohm resistor as the bottom leg of a voltage divider and this sensor as the top leg, I would predict you would see a range of approximately 0V with no pressure applied and 5 * 200 / (300e3 + 200) ~= 0.003. If you juiced up the 200 ohms up to say 300kOhms, I would expect your upper end to go up to about 2.5V. The bigger the upper resistor gets the closer your full scale gets to 5V, but the slower your response time will get because it's like you're charging/discharging a capacitor through a large resistances. It's also not going to do anything to linearize the sensor output for you.
The inverting amplifier configuration they suggest is based on the principal that the current flowing "into" the negative terminal of the op amp through the sensor will be equal and opposite to the current flowing "into" the negative terminal of the op amp. That is to say:
(V_n - V_T) / R_sense = (V_out - V_n) / R_F and V_n is driven to GND by the op-amp (so that the positive and negative terminals are at equal potential) so:
V_out = -V_T * R_F / R_sense = -V_T * R_F * (1 / R_sense)
We know that 1 / R_sense = conductance of the sensor is roughly linear with respect to the applied pressure from the user's guide (i.e. Pressure ~ 1 / R_sense, or Pressure = a * (1/R_sense) + b for some a and b) so:
V_out = -V_T * R_F * [(Pressure - b) / a]
This has the benefit of giving you an output voltage that is linear with respect to what you are sensing. Additionally, you can determine a and b through calibration at two (or more) fixed values. Finally you can adjust the range and slope of the output voltage by tuning R_F and/or V_T. The challenge then is that you need V_T to be negative with respect to your op amp supply voltage.
One way to accomplish this is would be a 9-volt battery with the positive terminal of the battery connected to GND of your op-amp and the negative terminal being V_T (effectively at -9V). You will then want to choose R_F so that you get an appropriate range of V_out per the above formulas. In tuning, V_out must not exceed the op-amp supply (probably 5V) or the amp will clip as well. There are more sophisticated ways of generating a negative voltage, for example by using another op-amp as an inverting buffer (as illustrated by U2 in this circuit which creates a virtual GND at the midpoint of the supply voltage), but I think the simpler approach should work of you, others can chime in if they disagree.
Long story short, I think you should follow their recommended circuit using the op-amp. You can get a DIP package with four rail-to-rail amps in it like this one, so with two of those chips, a 9V battery, and some resistors I think you've got everything you need.