Arduino + Pressure sensor
You could use a single instrumentation amplifier (INA) for the task. The main selection criteria:


Low input offset voltage. As you want to amplify mV signals this should be as small as possible as it will be amplified as well and can bring in significant error. 
Low input bias current. If you sensor has high impedance this current will produce some voltage drop on it which will bring in additional errors.
As small low frequency noise as possible. Noise produced by the amplifier in the 0.1 Hz - 10 Hz range should be minimum as your sensor's output is in the same region. (Page 3 has a figure about the sensor's 0.1 Hz - 10 Hz noise.)


I found Analog Devices' AD8237 instrumentation amplifier great for such purposes. The above mentioned parameters are the following for this amplifier:


Input offset voltage: typically 30 uV, max 75 uV, drift is 0.3 uV/°C
Input bias current: typically 250 pA, max 650 pA
0.1 Hz - 10 Hz noise: 1.5 uV peak-to-peak


Example circuit:



It was made for a strain gauge application which also means low output (for example 0 -  5 mV) signals, just like in your case. Resistor values and supply voltages would be different in your case you should determine them according to your application and the amplifier's datasheet is recommended to be read.