Choosing a reel
Specific reels are not really important but you should stay away from plastic components. A heavy reel or a light reel only matters to the fisherman using it. As long as it has a functional drag, any reel will work just fine. Beginner fisherman do not need to spend a lot of money on a reel.
A fly reel has 2 primary very basic functions:
Hold your fly line.
Provide an adjustable drag system that can handle your fishing needs.
The main differences in fly reels are:
How heavy they are (weight of the actual reel) that you have to carry around and hold while you are casting.
Size of the round arbor that holds the line. Most fly reels for trout are mid range around the size to fits a 6 wt line. Larger fish like Steelhead / Stiped Bass / Salt Water Fish - use a “large arbor” to hold the longer line. Salt water reels are much more heavy duty with robust drag systems and no plastic components.
What to purchase
Your local fly shop will be able to discuss the specifics of each reel they have available.
For fresh water needs a size 5-6 reel will be fine. Your budget will dictate how much you spend but you are basically paying for how light the reel actually is for you to carry around, and the capability of the drag system. If you are a part time fisherman and pay little attention to the weight of the reel for your infrequent fishing trips, you could start looking for reels in the $50 range. You do not need to spend a lot of money on a reel. Save your money and put it toward a nice quality fly rod. The fly rod is important to be able to cast, not the reel.
For saltwater the reel matters much more. You need a piece of hardware that can take the stress and strain of a long hard battle with the fish streaming the line off the reel many times in just one fight. You need to spend a good amount for a good saltwater reel and make sure you rinse it well every evening coming off the salt.