What are some good brown trout flies for a slow moving stream in early spring?
Depends.
Two approaches, that can be combined as you go:
The "soup de jour":
-You can expect midges almost anywhere in the country from winter into summer -esp so on slow water.
-Baetis are often the first mayflies all across the country, followed closely in the East by Hendrickson, and "Blue Quill" (Paraletophlebia).
-"Grannom" caddis come right in there too, East and West.
-Stoneflies are active and begin to emerge too, but are less apt to be found in slow water.
By Water Type:
-Slow water (either stream sections or just discrete areas of a given pool) often requires a stealthier/more patient approach, smaller/more realistic flies and finer rig, while in more turbulent water you can often get closer to the fish, and get away with larger flies and a courser rig.
Factoring in Trout Behavior:
-Productive waters (LOTSA food), esp those heavily fished, tend to make trout harder to dupe. Less productive waters (more meager rations) trout tend to be less choosy and more willing to "sample".
I fished a stream last week (see pic below) in which the browns were midging in that slow pond-like water. Took heroic efforts at stealth and patience, but the trout were not choosy and, despite feeding on the only available food (#26 midges), were happy to take any old #16 dry on 5X. The stream is not terribly productive and the particular stretch is not fished often bc it is such a bear to approach and cast there.
So, what patterns specifically? Decide where your stream fits in the above categories, then go either realistic or generalized ("attractor"/low maintenance). If the former, Google the bugs to get an idea of patterns out there, or others may chime in with their favorites. The reason I didn't just list a bunch of patterns that work for me is bc ... it depends. Realize, trout are not looking for a specific fly pattern, but food. And that comes in context.