Hi Matt,
It looks a nice little river.
It you are only starting river fishing the best start would be to approach the river very quietly and watch for rising fish but stay low as your shadow will put them down.
IMHO the way to start off is using wet flies. Use cover(if you have any) and keep it on the downstream side of you. Make sure to crouch when casting and when you are moving stand well away from the bank.
Fish a team of 3 wetflies. I would start with a hares ear nymph or a pheasant tail nymph on the point, a spider on the middle dropper(black spider,silver spider,snipe and purple,etc). As for the top dropper its up to you, if the fish are rising you could try match the hatch. In the summer evenings a wet brown or silver sedge sitting in the surface film will often get you the best fish of the day. A sparcely dressed bibio is also a great top dropper.
Cast accross the river and try land the flies slightly upstream of you(someone else mentioned a 45 degree angle). Let your flies drift down with the current. Strike at movements on the surface where you think you flies are. Twitches on the leader are also an indication. You can vary the speed of the flies by mending the line up or downstream. Do this gently though as this will also put fish down. When the drift is over and the flies start to dangle in the current leave it a few seconds before retrieving with a figure of eight as fish often hit you as your flies rise up in the water.
Dont false cast too much, with a wet line one of two flicks should get enough line out on a river of that size. To much false casting will dry out your flies and can also put fish down.
If there are spots where you can wade its even better, dont cast a long line and flick your flies into likely spots. Stand tight to the bank to give yourself cover.
Good luck,
Eddie