Barlows are almost always fitted just like an eyepiece. The Barlow goes into the star diagonal and your eyepiece goes into the Barlow. If as in the case of your scope it comes with a 1.25" diagonal and you intend to use 1.25" eyepieces you get a 1.25" Barlow.
Threads really are not a factor, but for info the rear port on almost all SCTs has a male 2"x24 thread. Large SCTs with larger ports come with an adapter for 2"-24. To that goes an adapter called the visual back which can hold an eyepiece or a star diagonal. Some but not all Barlows have on the eyepiece end T-2 threads for attaching a camera in place of an eyepiece.
A 2x Barlow is a good choice for this scope. However, an f/10 SCT is by design a long focal length, narrow field of view telescope. A 2x Barlow gives you effectively a 3000mm focal length, an f/20 scope. The greater the effective focal length the narrower the field of view, the darker the image, and most important the more you will see the effects of turbulence.
I found that it was a rare night when I could use any EP below about 12mm. The atmosphere is such a pain. But a few times I was able to enjoy, moment by moment an 8mm EP and once just once with Saturn a 3.5mm EP. The moon is a bit different but I would still say 8mm, sometimes 6 and even 4.
That said I have 3 Barlows, two that came with used scopes and a GSO 2" 2x with a removable lens cell bought to try various spacings. One Barlow is rock bottom quality Meade shorty, the other a better quality Meade 4000. I did not to my surprise find that one Meade was better than the other.
Would I have bought a new 1.25" Barlow? Probably not, at least not for visual.
What I did buy and highly recommend is the Celestron f/63 reducer-corrector. As the name implies it both reduces and to a degree, corrects the field curvature that is inherent in a standard SCT. True you get some of this by buying a 30 or 40mm EP, but I don't find them comfortable to use and the reducer as I discovered is very useful if (as you would be wise to do) you transition to EAA,
You might want to read about the pros and cons of Barlows (and their more expensive relatives the Televue PowerMates and ES focal extenders) as well as about the limits of magnification and resolution.
Enjoy your new toy, the 6" is good scope.