Maybe the biggest mistake made when an angler hooks up with a trout is holding it too tightly. It's exciting. Our adrenaline is pumping. It's natural to keep too much tension on the line. But you have to avoid this - especially with bigger fish. If the trout wants to run, let it. Keep only slight tension on the line - like the drag from your reel. Just make sure you hold your rod tip UP! It won't be long before the trout tires. When it does, start bringing it in. It will take another run. Fine. Let it go again & repeat the process. It's runs will get shorter & shorter. Eventually it will be close enough to net or take by hand. The number one key to playing a trout successfully is to keep your cool & remain calm.
On the subject of line tension, too many anglers have the drag on their reel set way too high. You don't need all that tension on the line. Your reel's drag should be set to offer only a slight resistance to the trout.
When the trout makes it's run, follow it. Don't allow it to go to another section of the stream without you. If it runs down or up stream follow it, reeling line as you go, so as to keep steady pressure on the fish.
If you've hooked into a large trout that's out in the faster current & refusing to move, get below it & pull your rod towards the bank you're standing on. Get your rod parallel to the bank. Most of the time this will move the fish out of the faster current where it's difficult for you to have any control.
If your trout runs under a log or takes your line around a root or other obstacle let the line relax entirely. Take all tension off the fish & wait. Eventually the fish will think its troubles are over & gently swim out from the tangles, freeing your line with it. Then you can resume your fight when your line & trout are back out in open water. If the trout's not moving from it's cover for a very long time, give your line a few pumps & then quickly relax the line again. Sometimes this will frighten the trout out from the cover they've taken.
Maybe the biggest mistake anglers make that causes difficulty in landing a trout is the rod they use. In these days of lighter rods & lines you want to make sure you have enough rod to land a good trout. Make the sport be more about finding & fooling the fish, rather than the fight. Besides, if you tie into a really big trout on a very light rod you might get into a prolonged battle with the fish that could end up killing it unintentionally.