While using spinning gear instead of fly gear to catch trout will be less expensive, there is definitely some essential gear that you'll need, along with a few other things that you may want to consider.
Spinning Rod and Reel
Obviously, you'll need a decent spinning rod and reel to fish with. You can use that 30-year old seven foot long two-piecer that you found in grandpa's basement, but I prefer to use ultra-light gear when catching trout in streams. For starters, fishing ultra-light is more fun and more challenging (especially if you hook a lunker!) than a longer rod with 8- or 10- pound test. Also, the shorter rod is less likely to catch a tree in your back-cast and I've found their a bit more accurate than larger light- or medium- action rods.
If you're purchasing for the first time, buy two ultra-light action rod and reel combos with a rod length of five feet or less. Why two? I've had a number of instances that my reel will run short of line due to a string of bad luck. With these small reels, all it takes is two or three snags high in a tree or under a rock on the opposite bank and then you're headed home. Having a spare rod in the car is a good way to make sure that you're not leaving the stream early. (At least not for a reason other than having caught your legal limit!)
I've found that spinners are the lure that I've had the most consistent success with when fishing for trout in streams - even more so than live bait! When you head to the stream have at least five spinners to choose from - this will give you options if you're not having luck with a given color combination as well as keep you in stock should you lose a lure through an errant cast into the trees or get snagged on a submerged object. These can be from a Mepps Trouter Kit, or a selection of lures from Panther Martin, Rooster Tail or others. I'm personally a big fan of Panther Martin, but have also had good luck with Mepps.
I guess really that's all you really need - along with a fishing license - a rod and reel and some lures. But really for a good experience where you'll be comfortable and to increase your odds of catching some fish, there are a few other items you'll want.
Hip Boots or Waders
Having to always cast from shore can be a hassle, there are trees in the way and often much of the stream is inaccessible due to the tendency for trees and bushes to grow on the banks. And while you don't need hip boots or waders to get in the water, bear in mind that trout are most active when the water is between 57 and 63 degrees. You'll get pretty uncomfortable pretty quickly in your swimsuit at those temperatures!
Unless you have large pockets on your jacket (and it's not a warm day) or pants (and you're not wearing waders or hip boots), you'll need a vest to hold your lures, pliers, stringer and anything else (snacks?) you might want with you at the stream. And the loop on the back makes for a very good way to keep your net close without it getting in the way.
If you're fishing from shore, you'll only need a net if you hook a big one that you can't pull out of the water by the line. But once you're in the water, you'll want to be able to secure your catch while you get the hook out and get it in your creel or on your stringer. Make sure you get one with a loop attached to the handle so you can secure it to your vest.
The hooks on a small 1/16- or 1/8- ounce lure are tiny, sharp, and pretty close together (especially those treble hooks on your spinners!), so buy a pair of needle-nose pliers to save your fingers from a nasty puncture when removing the hook. You'll be able to get that hook out much more quickly without having to worry about those barbs sinking into your flesh!
Some fisherman like the cool look of a fancy creel to store their catch, but they can get pretty gross inside, so I use a basic stringer to store my catch before heading home. I just have a bucket or pan in my car to lay the catch in to keep my upholstery fish smell-free. This is, of course, totally unnecessary if you plan to catch-and-release.
That about sums up the necessary gear for trout fishing. There are a few other things I'm sure I'll touch on in future posts, but for your first trip to the store (or just Amazon).