Was going to say Select and Mask as well but norman beat me to it.
Also, Yogi is adorable.
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Start with as high a resolution image as you have available (much higher than the one you posted).
Go to select and mask and use the quick mask tool to get the basic selection
Switch to refine edge and with a very small brush 10-20px at 1% spacing and 100% hardness work round the edges only
Switch to the plain brush and tidy up any final areas
Do not check decontaminate colours but do output to New Layer with Layer mask
To decontaminate add a new layer and switch blending mode to color. Clip this layer to the masked layer by Alt clicking on the border between the two layers in the layers panel. Brush lightly with color from around the edge to remove color fringes
When you work in select and mask and remove fringes, work with the new background in place. That way you only do as much mask work as needed to make the image sit in the new background.
Hate to disagree with everyone, but the way to do this seamlessly is to create custom brushes. Basically you would outline the dog , who incidentally is seriously cute, with the pen tool and create a layer mask. Then you'd draw onto the layer mask with the brush to mimic the direction and flow of the fur-sounds hard but is not. Afraid I don't have time this evening for a demo but will try and provide one tomorrow.
Hate to disagree with everyone....
Now it's never bothered you before Terri
I agree that brushes can be useful with hair or fur - although I tend to keep them to the end to finish off any areas that need it , once the image is on the new background. If the new background is to be a real image rather than a plain color it can hide a multitude of issues - and avoid a lot of unnecessary masking and painting to make it perfect on black or white.