The technique that you've described (downscale + blur passes) is pretty standard for achieving the effect that you want. Could you perhaps explain what looks bad about your current implementation? It's possible that you have a simple bug that you could fix, or that some simple tweaks could give you better results. If you could get some screenshots or a video, that would probably be helpful.

Are you using bilinear filtering when you downscale and upscale? It looks to me like you downsampled to 1/4 size in each dimension with point filtering, which is giving you that "blocky" look due to the aliased edges. So to start of I would make sure that you're using bilinear filtering for your sampler states when you're performing the scaling passes. 

It also looks like your blur is pretty narrow, which isn't helping you to get that "glowy" look that you want. I assume that those kernel weights are from a Gaussian function, and you're applying them to the 9 texels in the neighborhood of the center pixel that you're processing? That should give you as decent amount of blurring, but you may want to tweak the sigma parameter of the Gaussian to get a wider-looking kernel. It also looks like your weights aren't integrating to 1, which is why the resulting image is darker than the source image. You can correct for that by summing up your weights, and then multiplying your result by 1 / sumOfWeights.

You may also want to read through this article, which has a good overview of how to do an efficient blur for a bloom/glow effect.

On 7/2/2018 at 4:19 PM, Eric F. said:

I re-read the Rastertek tutorial this morning and they simply set the Alpha value of the pixel shader resulting color to 1.0f. This seems fine for their demo, but if I want to merge the blurred texture on my scene, should I simply leave the alpha as is at the exit of the shader?

That depends on how you combine the blurred texture with your scene. The two simplest ways to do that are to combine in a pixel shader, or use alpha blending. Doing it in a shader is more flexible, but requires an additional render target that will contain the result of combining the two images. Doing it with alpha blending is less flexible (you have to use fixed-function blend equations), but allows you to modify the "main" render target in-place. Even if you use hardware alpha blending, you may not need the alpha value at all. The typical alpha blend equation looks like this:
 

(SrcColor.rgb * SrcColor.a) + (DstColor.rgb * (1 - SrcColor.a))

Where "SrcColor" is what comes out of your pixel shader, and DstColor is what's in your render target. This is basically doing a linear blend of SrcColor and DstColor, similar to doing lerp(SrcColor.rgb, DstColor.rgb, SrcColor.a) in a shader. But that's usually not what you want for a glow effect, which is usually composited with a straightforward additive operation. In that case you don't need to use the alpha value in your blend equation at all, so it won't matter. You may still want to multiply your glow texture by some scale factor before combining it with your scene, but you can do that very easily by just multiplying it with your pixel color in the shader.