For mountain rivers with strong currents, your best bet is a fly rod between 8 and 9 feet with fast or medium-fast action. That range lets you control your cast well in tight spaces (which you usually find between bushes) and keep constant tension against the current. If you prefer something more versatile and budget-friendly, a 6 to 6.5 foot spinning rod with medium-high power also works great, especially if you use small lures or light spinnerbaits.
As for brands with a good balance between quality and price, there are solid options without breaking the bank. Don't go chasing luxury names; mid-range rods from established manufacturers are usually more than enough to get started. What's really worth spending a bit more on is the reel: a quality reel transmits sensitivity better and lasts way longer. The rod material matters less than its balance and how it feels in your hand.
One practical tip: try out several before you buy if you can. Each one has a different feel, and a rod you actually like will make you enjoy your fishing trips so much more. In cold mountain rivers you're going to spend long stretches with nothing biting, so having gear that's comfortable makes all the difference.