Electronic – Cannot find GND on 7 Segment Display Datasheet

You are not using enough voltage.
The page you referenced notes that there multiple LEDs in series.
They say:

• Large (58mm) 7-segment LED display
  Digits color can be either red or green.
  Type: 12101BEG
  If max: 30mA
  If peak 1/10 dutycycle, 0.1ms: 100mA
  Vf red: 7.2-8.8V
  Vf green: 8.8-10.0V
  Each segment consists of four LEDs connected in series.

In most cases simply trying every combination in turn will work.
There is a small chance that you will destroy segment LEDs with reverse voltage.

Supply two leads from psu via a resistor so you can easily swap both leads.
Try each combination in turn.
A1 A2 A3 A4 A5 B1 B2 .......

The red wire is hooked up to the 5v output on the arduino and the blue to the ground. As can be seen the decimal point has been lit up in green, and if I move the red wire four pins to the left the decimal point turns red instead.

As they say CA = common Anode = common +ve you may have
2 x Anode's
+ 8 x Cathodes.
So Bottom right may be common green anode and
bottom left = common red anode.

SO increase Vsupply to say 8V, use a 10k resistor for initial safety, put V+ on bottom right and try all pins with v-. If that works then put V+ via 10l on bottom left and repeat.

The supplier should be able to provide a datasheet.

Life can be tough. CMOS ICs from the CD4000 series, like the CD4026 can only source and sink very little current, typically 1 mA at 5 V and that will be much too little for a typical 7-segments display. So I'm afraid you'll need the transistors, especially since you're now already asking for more brightness.

But You don't necessarily need all these discrete components. A ULN2803 replaces 8 transistors, so you'll need 1 IC per display. Important note: I just read that your displays are common cathode. The ULN contains an array of NPN transistors, switching to ground, so they can only be used with common anode displays.

edit
starblue found a high side version of the ULN2803 in the UDN2981, so this is suitable for common cathode displays. I don't know about availability; Digikey lists only two versions, both from Allegro, as as non-stock, call for price.
Inputs are active-high, so it can be driven by the same devices which would control NPN transistors or the ULN2803.
end of edit

Alternatively you may use low-power Schottky ICs, but then you'll need the counter and the display decoder separately; I don't think the combination like the 4026 exists in LS-TTL. The 74LS90 is a decimal counter, and the 74LS247 a 7-segments decoder, which can drive LEDs directly. This is also an active low output IC, which means common anode:-(.

Other solutions you've seen requiring less transistors may have used a microcontroller and a multiplexed display. Then for 6 digits you'd only need 6 + 7 transistors, instead of 6 \$\times\$ 7.