I'll use your initial scope choice to explain this.
3x9 x40 - I have this exact type of scope, albeit made by Burris instead of Leupold, on both my slug guns, my .30-06, and my muzzleloader.
the bottom level of the scope's magnification is 3 power. Meaning if you cranked the scope all the way down, this would look 3 x closer.
The maximum magnification is 9 power. That means if you cranked the scope as high as it goes, things would appear 9 times closer than they are.
The 40, which stands for 40mm, describes the diameter of the bell (i.e., the front lens of the scope). The bell is 40mm in diameter. The larger the bell, the more light it will let in and the better your view will be in low light shooting conditions (i.e., dawn and dusk).
On my long range rifles (i.e., .270 and .300 Win), I have Burris Signature Series scopes that are 4-12x with 52mm bells. On my varmint gun (i.e., .220 Swift) I have a Burris Signature Series scope with a 8-32x with a 52mm bell.
If I really had the money when I purchased those rifles and scopes, I would have opted for the 30mm tubes with the Burris Black Diamonds, but those scopes run around $600.
To answer your question, the scope your are thinking about getting (i.e., the 3-9x 40mm) will do just fine for a beginner. When I started out deer hunting, I was using a straight 4x scope with a 32mm bell and I killed my first deer within 15 minutes of my deer hunting career. Trust me, it hasn't been like that all the time.
At the end of the day, but the best scope you can afford because it does make a difference, but you do not have to get a loan/mortgage for your glass and to be able to hunt.
I know guys that hunt without scopes and they do just fine.
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