This is Photoshop. There is no one way to do anything. And there are a good dozen ways to extract an image. I agree with Abdul, zooming in after you use your basic tool is of utmost importance and it sure takes practice.
I've never tried the blur tool for this but I can see where it would work well in some situations. This is one of the things that takes time to learn: what tool is best for what kind of image.
There is defringing to remove excess pixels layer>matting>there are a few choices here, there is refine selection or refine mask, there is painting on a mask, by hand of course, or doing the same thing with eraser (but that is destructive so I prefer masks). In some cases, there is just going to be some time that must be devoted. If you have a busy background and a soft-edged, maybe furry for instance, subject, you're going to have to do some cleanup with a manual method. Of course, how picky you want to be depends on the use of the subject. Also, if you remove a BG that is really dark colored and it's going to be placed on black or other dark colors, you needn't be so precise. The more precision you can learn, the better.
So, here are my favorite selection tools - and different artists have their own preferred methods.
Pen tool with mask or selection cleanup, or both. This takes learning and teaching but you can learn much from a tutorial and Hawkeye gave you a quick tut on it (not enough IMO - no offense intended Hawkeye) but a start and the concept). I have a few tutorials on the pen tool if you're interestd.
Quick select using pretty much the same as above. I like to use a really large brush on rounded shapes that are easy to access as then the brush and the pixel removal is easy to control. And in less-defined area smaller brushes of varying hardness and opacity. The wand has some similar qualities but I find the newer quick select to be a better tool.
Channel masking. This is too intense to describe (I could if I wanted to create a new thread, lol) but can be very useful if you have a solid BG that has the opposite tonality of your subject. There are a lot of tutorials on the web for this. If you want I have bookmarked about 5 of them.
So, those are my favorite. Under that matting selection, there is remove white or black for defringe type choices and I imagine erasing a BG of W or B, believe or not, I've never used it ... or the magic eraser. I think both are probably pretty messy if you want a clean selection.
Well, that's my synopsis. I'm sure you'll have lots of questions, but I will repeat here, you must learn the basics and you must learn them in a logical order. You can't jump ahead and become a PS expert or even apprentice or whatever you care to call it.
