The main focus of this will be river trout but there's no reason it can't apply to other species. It goes without saying that the best angle of approach is the one that lets you deliver the fly the way you want to. It could be your casting options are limited due to the lie of the land or a particular wind or it could be every option under the sun is available. There's three main options though - upstream, downstream and finally across.
Probably the most common approach for dry fly but it's equally useful for nymphing and even fishing streamers. Dead drifting a bunny pattern can be incredibly effective on certain fish! I typically fish up if I'm prospecting the water (which I rarely do these days) or if the water is crystal clear. It's by far and away the best approach for spooky fish when you've got nothing to cover behind as you can approach from the fish's blind spot. It also lends itself to a nice drift most of the time and it's easy to manage your line however your options for mending during the actual drift are pretty minimal. It does have a few drawback however - it's very easy to plonk the end of the fly line too close to the fish from an overshot cast (we don't do that though, right?), you always need to take some sort of angle in towards the fish so you don't line it and the leader is always going to lead the fly into the fish's window. Whether this final point makes a difference or not is up for debate. I think it does though on particularly difficult fish.
Perhaps the grand-daddy option for a good drift. With the right cast and a good amount of slack you can get some truly wonderful drifts approaching from directly above the fish. In fact, on the canal in Croatia that you'll head Paul talk about that's exactly the way to get them. It was early doors when I fished it with Paul, both being new to it and sure we picked up a couple fish by fishing down but we didn't do well. Paul's now picking up good fish by crawling down the banks with a very targetted downstream approach. And there's the rub - the bloody fish can see you. If you're going downstream here's a few pointers 1) be James Bond level stealthy, 2) watch that slack! You need it but if you've got too much you're going to strike into thin air when the fish takes, 3) when the fish takes, allow it to turn and go back down before striking or you'll pull the fly straight out it's mouth. The cool thing about downstream fishing is you can bring that fly to life by tightening the line/slackening it again, make it dance, make a mouse run, make a nymph lift, make a sedge skate. There's so many options.
For me this is where things get very interesting. I'm thinking here of a single dry or perhaps a single nymph. You've locked on to where a feeding fish is and my God you don't want to fuck it up. Well, get your backside across from it, get yourself slightly upstream of it and be prepared to make a wonder cast that's going to result in the perfect drift. You want to aim 4-5 feet up from the fish, send the cast and while it's in the air make an upstream reach. The end result is a cast aligned with the fish, with a great drift thanks to some slack in the leader (you will of course have set up your leader to collapse nicely into little S shapes), the line positioned both upstream of the fish and with appropriate slack to not cause drag AND the fly is going to be the first thing into the fishe's window, not the leader. The difference this approach made to my own fishing was mind blowing. All of a sudden those imopossible fish feeding away hard in amongst thousands of micro surface currents that would refuse time and time again when fishing upstream would be scoffing down my offerings. I've got to thank a friend of mine Jim Lees for that one, he switched me onto it as about the only way to properly combat drag on this particular river. It changed my approach full stop. Thanks Jim! He's also a mean fly tier so check out his stuff here - http://www.palewatery.com/blog/
There's a few more general things to consider also like the wind direction, the sun position (for shadows and flash) and the currents themselves. Sometimes these simple guides above just won't cut it and you'll need 14 aerial mends all at different points in the line. So there's another good point, brush up your casting, there's plenty on mends here - http://www.sexyloops.com/flycasting/tbasic1.shtml
The final point to bear in mind is that regardless of what tradition says whether it be you fish drys upstream and wets downstream, exercise a bit of common sense and do whatever you want that will get the desired result. Hell fish some wets upstream or start dead drifting streamers. There are no rules in this game.