Well there are lots of good rods and reels out there.
I usually suggest starting with a 6'6 to 7 foot medium spinning rod as the best all around bass rig. It can handle most of the typical finesse baits line Sencos, grubs, tubes, lightweight texas rigged plastics etc. I would match that with a good spinning reel like a Pflueger President, President XT, or Supreme in a size 30 or 35.
But that combo will not do what you are asking to do. I fish with a dozen baitcasters and 3 spinning rods. My wife uses only spinning or fly fishing gear. Finding the spinning rods that have virtually the same action as some of my rods has been a struggle. I now have 6 rods in her arsenal and each one does exactly what I was looking for.
Her frog rod would be a perfect choice for your needs. I purchased her a Dobyns Sierra SA 705SF. That is a 7 foot spinning rod listed as a Heavy power rod. It has a soft but fast tip, exactly like the action on my Dobyns Champion DC 705C rod I fish frogs on. I can throw a Strike King Popping Perch, or a typical hollow bodied frog, a very very far distance, yet it has the right tip to work it correctly and a ton of power to pull any bass out of a junk yard. The rod is lightweight and very well balanced. It costs $170.00, so that fits in your budget also. Gary Dobyns' list of suggested lures for this rod includes Carolina rigs/Frogs/Jigs and as secondary suggested techniques Buzzbaits/Spinnerbaits/and Walking Top Water Baits.
Now for a reel there are a bunch of manufacturers to choose from. I prefer to choose my reels for longevity, quality and smoothness. Most of my spinning reels are Pfluegers. We matched her rod up with a Pflueger Supreme 35. This reel was chosen for several reasons, it is made very well, its smooth, and it also has a high speed retrieve. It retrieves the same amount of line
( inches per turn of the handle) as my 8-1 Daiwa Tatula baitcaster. For frogs that higher ratio means you can retrieve a lot of line fast. If a blow up and miss occurs you are ready for your next cast quickly. This reel will run you $100. This reel is spooled like all of my reels. I has half of a spool of Strem 14 pound Clear Blue mono backing and then it is filled up with 30 pound Power Pro Spectra braid. It is 30 pound test but equal in diameter to 8 pound mono. It cuts through Lily pads and other vegetation easily.
Since this is her exact combo I know it balances well, and is a joy to fish with, I am sure you will be happy using it.
When you are ready for your next setup take a serious look at my first suggestion of the Medium action rod. Instead,if you decide you want the perfect rod for throwing jigs and texas rigs, then look at the Irod Genesis II IRG 713S rod. That is her jigs rod and again I would match it with a either a President or Supreme reel in say a 30 or 35 series.