I've been fly fishing for 47 years and have never thought of fly fishing as an elitist sport. There have always been rods, reels, lines on the market to satisfy any budget. One doesn't need to spend $700.00 on a rod, or $700.00 on a reel, or even $50.00 on a single-hand fly line to get a good serviceable fly fishing outfit. There are some very nice rod, reel, line combos on the market for under $200.00.
That said, you are right in thinking the "Walmart special" you got is not a very good rod; however, if that is what you can afford at the moment, use it until you can get something better. After you can affort to get a better rod, a look at the beginning fly fisher combos (they have rod, line, and leader included) would be the best way for you to go. There are quite a few very good, serviceable beginning fly fishing outfits on the market for between $125.00-$200.00 that would serve you well for many years. Some of the manufacturers of these good cheaper outfits are St. Croix, Echo, Redington, G.Loomis, and Lamiglas. Trust me, you won't need a high end $600.00-$800.00 rod any time soon, if ever (even though that is what I use).
As far as the leader goes, the best thing for you would be the tapered leaders of 7 1/2' with a tippet that is 3X (about 6# test). This is strong enough to land any of the fish you are targeting at present and is not too large a diameter that would put the fish off. While you are at it, buy an extra 3X tippet leader spool or two to replace the tippet on the tapered leader, which will get shorter as you tie on and take off flies. To tie the tippet on to the leader (don't do this until you have used up about 18" of the leader), simply tie a double overhand knot with the new tippet section (about 18"-24") overlapping the leader (in other words, you are going to pass both the end of the leader and the length of the tippet through the overhand loop twice). Put a little spit on it, and pull it tight while holding on to all four pieces of it. Very simple knot (you may already know how to tie this since you have been using spinning and casting gear) that is very strong.
There are fly lines which are made specifically to help new fly casters cast better, and the best part is they are cheaper than the onces made for more experienced casters. Scientific Anglers (SA), Cortland, and RIO all make them and they sell for around $25.00 give or take a few dollars. Once of these in a WF7 would be proper for your 6/7 rod. You are thinking correctly as regards the WF line being a better choice than the DT. The WF floating line is far easier to cast over 30' than the DT.
Backing is inexpensive and you really ought to have some on your reels. For the fishing you are currently doing, 20# dacron backing would be a good choice. You should have between 50 and 100 yds of it underneath you fly line so that you could allow a larger fish to take out your line and not worry about running out of fly line. This will help you land more fish. I'd never fish without backing on my reel, even if only fishing for trout under 6" is small mountain brooks.
As far as flies, you pretty much get what you pay for. There is a huge difference between the 55c Walmart flies and the 2/$2.00 ones. The hooks are better, they are tied better (meaning they will stay together for more fish), and are better proportioned.
Hope this has helped provide useful information to you.