AD8628 Op Amp Non-Inverting Amplifier Not Providing Expected Gain with 0-3V Rail

I know this is probably an embarrassingly simple question but I'm at a roadblock. I'm an aerospace engineer working on a side project privately to gather data from a strain gauge using a NI DAQ. The output is 0-36 mVDC so I need a gain of 3 to better utilize the DAQ.  My EE experience is limited to a couple EE classes I took during my bachelor's. I have an AD8628 op amp and built a basic, non-inverting negative feedback circuit, as shown in the attached picture. The rail voltage is 0-3VDC. For the life of me, I cannot understand why it is not working. It does not not seem to have any gain, but instead the output voltage is actually lower than the input. I'm sure I must have wired it incorrectly but i cannot tell how. Any help would be greatly appreciated.

A strain gage is a passive device. It need an excitation. Most strain gages are incorporated into a bridge circuit that then needs a power supply. Because you sould be picking up the voltage across the bridge, you'd need a differential amplifier configuration or an instrumentation amp.
NI has strain gage signal conditioners, but you'll pay $$$ for one.
Check out this article.https://www.edn.com/design/test-and-measurement/4384454/Tips-for-better-strain-gage-measurements

This article explains how strain gages work and tells you about the conditioning circuits you might need.https://www.dataforth.com/strain-gage-signal-conditioner.aspx

Martin is right, a strain gage has two outputs neither of which is connected to ground.Here's a diagram from Martin's Dataforth reference (which is very good:
Points 1 and 2 are your excitation, so neither output will be at ground potential.What you need is a differential amplifier that will accept both inputs.I would suggest you go through the op-amp tutorials on our sister site Electronics Tutorials:https://www.electronics-tutorials.ws/opamp/opamp_1.html
No.5 deals with differential amps.  You'll probably be able to skim thorough a lot of it on the way.  The diff-amp tutorial specifically mentions a wheatstone bridge amp which is essentially what you are trying to do.It sounds like your strain gage is basic with no inbuilt amplification, so this may be your problem.

You don't say what sort of  NI DAQ device you are using. Is it differential or single ended DAQ input, what are its input specifications and resolution.When you say strain gauge, do you mean a commercial load cell, or a strain gauge foil.If you are applying a foil to a surface, this is a complex procedure, and foils are usually only designed for the thermal expansion coefficients of steel and aluminium.From your comment I am assuming a commercial load cell, the output is normally expressed as mV output/V of drive/ Kg.This allows you to compute the gain required  for the  optimum dynamic range.For best results, ratiometric  DAQ measurements are desirable, this requires access to a buffered ADC reference  voltage to drive the load cell, otherwise  a stable load cell drive voltage  will be needed commensurate with precision you require.What sort of stability(temperature range in particular), accuracy, and resolution do you require.How far is the transducer from your DAQ, and is the environment electrically noisy.Do you know the maximum load the device will experience, since for best results you need to match the load cell to the load for best dynamic range. They normally have  a  safety capacity 50% over the rated load.There is a note here that may help, as an engineer you should be able to follow most of it, even if the subject matter is new to you. There is another here that is more general , but an excellent  one none the less.Cheers,Richard

Fair point, I didn't explicitly say the strain gauge was powered by a 12VDC excitation voltage, but it is. I already have the basic output from the strain gage, 0-36mV, which I'm able to read and record on my NI 6289 DAQ,  which incorporates signal conditioning.  My problem is the lowest range the DAQ can accept is -100 mV to +100mV. I'm simply trying to increase the output of the strain gauge by 3(so that the output will be roughly 0 to +100mV) in order better utilize the available resolution of the recording capabilities of the DAQ.

 im using a NI 6289 DAQ. I'm utilizing a differential input. The DAQ is capable of 18 bit resolution but the lowest input range is -100mV to +100mV. It is a commercial load cell. Given the 12V excitation voltage, the strain gauge output range is 0 to 36mV (3mV/V) so i can't really take full advantage of the available resolution capability of the DAQ since the load cell output range is only about 16% of the lowest input range. I am trying to increase the output range of the strain guage by creating an op amp  circuit with a gain of 3 so I can at least use nearly 50% of the lowest input range. I'm utilizing a AD8628 since, from what I've read, it has very little noise and is designed for precision DC applications like strain guage. 
Temperature stability shouldn't be an issue since the test environment will be quite consistent. The load cell will only be about 1 ft from the DAQ and the environment will not be noisy. 
Really I'm just trying to increase the output of the load cell because the lowest range of my DAQ isn't low enough.
I do appreciate your help.

the steps are given by profesional electricianThere some basic steps that give you help to wiring onAmp 
Step 1: Attach power supply positive, negative, and groundterminals to a breadboard.Step 2: Connect terminals to breadboard nodes.Step 3: Install Op Amp.Step 4: Attach Power to Op Amp.Step 5: Wire Input Signal from Function Generator.Step 6: Inserting Resistors.Step 7: Wire Oscilloscope to Circuit.Step 8: Powering on the circuit and setting up a testing experiment.Thank you.

Scott...beware... >the strain gauge output range is 0 to 36mV (3mV/V) ...> op amp  circuit with a gain of 3 so I can at least use nearly 50% of the lowest input rangeDepending which way round you connect your strain gauge, you will get an output of 0 to 108 or 0 t0 -108 mV, either of which will exceed the +/- 100 mV input range of your DAC.  Either reduce your gain or reduce the excitation voltage to bring it within limits.  You could offset the output to go from - to +, but that adds extra complications.

Are you using the load cell close to capacity, and do you need the full 18 bits. If the answer is yes to the first, and no to the second, then you probably don't need an amplifier at all. With a 12V load cell drive, your common mode voltage is 6V, which is fine for the ADC. However, see  the comment about load cell drive voltage below.You need to work through an error budget, to figure out what stability you need for your front end given the number of bits required. 18 bits is around 4ppm, if your design is not good, a degree temperature change has a significant effect, you need to carefully consider that. How stable and noisy  is your 12V supply to the load cell, since the load cell output is related to drive voltage. As I said, it is common to use a ratiometric measurement, where the ADC reference is used to drive the load cell. If you can't do that,  you need  a load cell drive voltage that has the performance to match the bits required from the ADC. It is common practice to use another channel of the ADC to measure the load cell drive voltage, but obviously it has to be within the input voltage range of the ADC, which is 10V. It is also common to use fully differential input and output op amps to drive ADC's, they are especially designed for this sort of application,  the THS4561 works at 12V. You can set the gain with precision resistors. The performance, offset voltage drift for example, is nowhere like that of the AD8628, but they are designed for use with 16 to 24 bit ADCs. Note that the AD8628 is only as good as the stability of the components you place around it.You need to more fully specify your requirements, since this determines the number of bits you actually need, and therefore the precision and stability needed for the design of the front end conditioning. I use load cells, both custom and commercial in a lot of projects, I am always careful to only get the resolution and stability I really need, since each extra bit requires  more effort for it to be meaningful. I often  use 24bit Sigma Delta ADC's, which generally avoids the complications of adding front end conditioning, but it still requires care to get the performance specified in the data sheets,cheers,Richard

My plan to avoid this has been to not use a particular load cell near its maximum capacity and instead utilitize a higher capacity load cell to capture those data points. I do understand this will can result is a bit of error given that load cell error is dependent on the ratio of measured load to maximum capacity, but it will be acceptable.

Sounds like you are well across it Scott - a differential amp should sort your problems out.  Let us know how you go.

Are you using the load cell close to capacity, and do you need the full 18 bits. If the answer is yes to the first, and no to the second, then you probably don't need an amplifier at all. 
With a 12V load cell drive, your common mode voltage is 6V, which is fine for the ADC. However, see  the comment about load cell drive voltage below.
You need to work through an error budget, to figure out what stability you need for your front end given the number of bits required. 18 bits is around 4ppm, if your design is not good, a degree temperature change has a significant effect, you need to carefully consider that. How stable and noisy  is your 12V supply to the load cell, since the load cell output is related to drive voltage. As I said, it is common to use a ratiometric measurement, where the ADC reference is used to drive the load cell. If you can't do that,  you need  a load cell drive voltage that has the performance to match the bits required from the ADC. It is common practice to use another channel of the ADC to measure the load cell drive voltage, but obviously it has to be within the input voltage range of the ADC, which is 10V. 
It is also common to use fully differential input and output op amps to drive ADC's, they are especially designed for this sort of application,  the THS4561 works at 12V. You can set the gain with precision resistors. The performance, offset voltage drift for example, is nowhere like that of the AD8628, but they are designed for use with 16 to 24 bit ADCs. Note that the AD8628 is only as good as the stability of the components you place around it.
You need to more fully specify your requirements, since this determines the number of bits you actually need, and therefore the precision and stability needed for the design of the front end conditioning. I use load cells, both custom and commercial in a lot of projects, I am always careful to only get the resolution and stability I really need, since each extra bit requires  more effort for it to be meaningful.
I already have the basic output from the strain gage, 0-36mV, which I'm able to read and record on my NI 6289 DAQ, which incorporates signal conditioning. My problem is the lowest range the DAQ can accept is -100 mV to +100mV.