Despite what the purists try to say, dry fly is the easiest, most aesthetic and most foolproof method of catching river trout I know. Upstream wet and nymph, to my mind, require greater levels of skill/knowledge etc. as you can't see your fish (in the Tees anyway due to heavy peat staining) and take detection is more problematical. Unless the river is bombing through or it's unseasonably cold stick to dry fly. You may not catch as many as an experienced wetfly man but it's usually reliable for a novice.
Initially, I would to find some smoother water with a medium pace and depth - near the middle of a pool is a good starting point. Don't cast!! Just study the water upstream of your position and search for rising fish - there's nearly always some!
Mark where they are as acurately as you can and attempt to land your fly (single) 2-3 feet upstream of the rise form and try to let it drift down over the fish without any movement from you (retrieve) or the water (drag). You will need to think about all the slack line that builds up as the fly drifts down - so try to keep in touch with it without moving it. If your fly gets taken strike! Easy!! No need to look for subtle indications of a take like with wets or nymhs.
Don't spend too long in one spot, work your way upstream looking for the risers.
AT A PINCH stillwater tackle will catch but better to have something lighter (I like a 9 foot 5 weight) but don't go too light as wind then makes things more unpredictable. I like a double taper line but a weight forward line is OK too. A floater is all you need and the reel is just a container for line on a river, so, if needed save money here. Get a couple of tapered leaders about 4x and tie a small loop in the thin end. You can then tie on 3 foot or so of 21/2 - 3 lb mono loop to loop as a tippet (easy to change and works for me). I like Bayer Perlon from the coarse tackle shop.
No need to get hung up on flies too much - trout are pretty dumb and accomodating fish! Stick to size 16 - 14 to start with. Grey ones, black ones and brown ones will do for starters. Patterns to consider are Klinkhammers (small - some are huge for the quoted size) standard hackled drys and sedges/black gnats etc. etc.
Some floatant and something to dry your fly on (kitchen towel or chamois leather) and you are off playing the worlds greatest game - river dry fly fishing for wild trout!!!
Good luck and let us know how you get on...