Binoculars with 7-10X magnification are fine out to 700-800 yards for general viewing and big game identification. Viewing at 2-3 miles can be an entirely different thing. If you just to want a closer look at the mountains, binoculars should be fine. If you really want to have a good look and be able to spot game animals at long range, you’ll want a good spotting scope and tripod.
With a proposed budget of $300, I’d look for a pair of binoculars and forget the spotting scope for now. I recommend something with 40-42mm objective lenses. I’d also suggest buying used, provided you go with a manufacturer who offers excellent warranty coverage and good customer service.
I’d start by visiting a local sporting goods store and comparing all the binoculars they have, even those above your price range. Cabela’s would be an excellent choice, if you have one nearby. Figure out what you want, then watch the online auction sites.
In your price range, I can personally recommend the older Bausch & Lomb Discoverer 7x42’s that were made in Japan. Unfortunately, they’re long discontinued and you won’t find them in stores. Later versions of the Discoverer binoculars were not made in Japan and aren’t the same, so you need to be careful. They provide a crisp sharp image, are waterproof/fog proof, rugged and have excellent eye relief. They’re my go to binoculars for Michigan deer hunting, even though I have more expensive optics to choose from.
I have a pair of Swarovski 10x42 SLC’s that were rebuilt by Swarovski USA after over 10 years of hard use by the previous owner. I just wanted the rubber armor replaced, but they completely went through them and replaced the prisms and hinge, the lenses and the armor, at no cost to me. They came back looking like a brand new pair! I later picked up a nice pair of Leica Trinovid 10x42’s at an estate sale, but ended up selling them. They were every bit as good as the Swarovski’s, but Swarovski has much better warranty coverage (25 years, no questions asked).
For western hunting, I use a pair of Leica Geovid 7x42 BD laser rangefinder binoculars that I also bought used. They’re an early generation and are fairly big and heavy, but it’s much more convenient than using standard binoculars and a separate laser rangefinder. They have superb optics and range out to 1300 yards, which is way more than I need.
As others have mentioned, viewing with quality optics will spoil you; especially if you do it for any length of time. You can take some of the sting out by purchasing used, bet there’s a reason people spend between $1000 and $2500 for a good pair of binoculars.
I’ll save my thoughts on spotting scopes and tripods for another time.
