For all around trout fishing, and the streams you describe, a nine foot rod designed to handle a 5 weight line would be a great all around set up. (A long rod is helpful for mending line in current to get a natural drift with the fly.)
If you’re new to fly fishing, keep in mind a good fly line will run 30-70 bucks, plus 7 bucks for backing for the fly reel. If you’re looking to keep it within 100-150 all in (except flies) you might want to browse some past threads on this forum to see what others recommend (lots of options) in terms of complete outfits.
I'm sure others will weigh in with other good ones, but LL Bean has 2 good ones in your price range to look at:
Their Quest 2 series outfit runs 99 bucks for a 2 piece rod (plus composite plastic reel, fly line, backing, leader and tube)
And their StreamLight outfit with a bit better blank and components and better reel. The 2 piece rod outfit goes for 169 for an 8’ 6”rod or 179 for a 9’ rod (four piece rods are more convenient for air travel but run higher). These outfits also include rod tube, reel, backing, fly line and leader.
You can find links for both here:
Fly Rod and Reel Outfits: Fishing at L.L.Bean
There are also a lot of other options out there for outfits from big name online guys, and you can also put a decent outfit together for a bit more by mixing components from different companies.
Another way to go might be to put some trust and a bit more money (depending on what you end up with) into your local fly shop, and have them put together something within your budget. The advantage of this is that they’ll have an investment in getting you off to a good start, and will maybe throw in a few informal casting lessons or a few freebies and otherwise really help you out with the learning curve. Getting help with casting is a HUGE plus for a beginner.
The downside might be a more limited selection of different price points of gear, but typically the prices for the same brand/model are very close if not identical to the big box stores. The local shops, which used to be the backbone of the sport are a great resource, but are having a tough time competing with the big boys. But they should have a range of inexpensive, decent models to choose from that you can wave around out back and cast a few, and can put together a great outfit for you to stay in your budget. If you were already anticipating the cost of the fly line, backing etc, in addition to the 100-150 figure you’d be at around 150-200 for a complete outfit with line and backing and it makes more sense to look at individual components anyway.
If you let folks know where you are (in your profile) you’ll probably get some recommendations for fly shops to check out.
Hope this helps.