My style of play seems to have settled into a steady method, so I thought I'd describe it here. I hope you find it helpful. Most of the puzzles we're set respond well to this set of techniques, with an initial rush of points, then a series of increasingly tougher "walls", where you just can't get the score to go higher, but you eventually find a way through.
Very Short Summary
(for those with no time :) ) Wiggle/shake/nudge till it doesn't work any more. Rotate/straighten a couple of helices, then move on to walking the backbone doing local wiggles, and again doing sidechains. Repeat this several times: all local wiggles till it stops, then sidechains, and back again. Then in the endgame stick with small pulls and local wiggles and you're done.
Opening Moves (Note: Following a suggestion from a team-mate, I've it is worth repeating the opening a few times, varying it slightly each time to see where you end up. One suggestion is to interrupt the initial long wiggle at some point, shaking the sidechains, then resume the wiggle. Try interrupting at different times, nudging before resuming the wiggle and other such combinations. Mix it up, see what happens.)
Wiggle, then Shake: Start with a global wiggle (Press W). The puzzle is usually configured so that this is best opening move. When it has run as far as it can, do a global shake (Press S). Repeat this pair of commands until no more points are gained. This set will probably work one or two times.
Nudge then Wiggle: We now need to perturb/kick/hit the structure and knock some more points out of it. Pick any point you like and do a small drag on it, preferably pulling it towards the center of the puzzle. Don't drag it too far; losing a couple of hundred points is about as high as you want to go, though larger nudges may also work sometimes. Global wiggle after each nudge and see if it yields more points. Start using Restore-Best to make sure you're always working from the highest score. This will probably work at least two or three times, but you can continue using it throughout the whole puzzle.
If your score ends up really, really far below other players, especially on a new puzzle, consider resetting and trying a different combination of opening moves. Delay using the Shake commands as long as possible, since they seem to "lock in" the current structure and may make it harder to move forward.
Small Rotates: The Tweak-Rotate tool has shown itself to be a smart shortcut to quick point gains, but I know only one way to use it productively. Find a helix (or make one if you don't have one) and give it the minimum number of loops on each side to work with. Rotate it by just ONE position. By this I mean when you click the purple pointer to make it move, stop it again after just one move, which takes less than a second. If you're feeling adventurous let it move two positions. Try a global wiggle and/or a local wiggle. This will sometimes yield a large jump in points for what seemed like such a small move, but I guess the system is very sensitive to rotation. I find the "straighten" command in Tweak works better than the rotates, but those are good too.
Variations on this will include unlocking more loops for the rotate to work on, and reducing the number of segments in the helix and trying the rotate again.
Sidechains : As of late July 2008 I've finally developed a basic understanding of sidechain positions and how important they are to reaching a good score quickly. The "Shake" command is great, but you must understand that it repositions all the sidechains to get the maximum score with the CURRENT backbone alignment. If you want to find a better one, walking the sidechains (see the link) is a good method, as is randomly picking sidechains to reposition, then wiggle (or nudge-wiggle) to see what happens. Early on in the game this will yield big point jumps fairly easily. Later on you'll probably have to walk the whole protein testing every sidechain to find one that will yield more points.
Local Wiggle Strategy: I'll use this a little bit, but since an August 2008 update it works less well than it used to. It's still worth trying, but don't spend as much time with it as before it changed.
Bands: Other people use this more productively than me, but sometimes instead of a Nudge-Wiggle, I'll use bands to try and reduce the overall size of the structure, or close a gap in its overall shape. Zoom out and consider the whole shape - if its looking like it wants to be a certain shape, help it become more that shape. If some part is sticking out, use bands to pull it in, then clear the bands, shake (to help lock it in) and global wiggle again. As I said, I don't do this as well as other players, so their techniques may be more helpful. Still, it sometimes gets my over a wall. If you use bands later in the game, small moves are better than large. Press W to start the wiggle and press it again almost immediately -- the move will be kept small.
Additionally, you should continue to use "Nudge then Wiggle", especially after any decent point gain. The structure may be ready to naturally re-adjust itself into a lot more points than you can create with tiny moves.
Most moves in this phase of the game will be less than a point, possibly even less than the resolution you can see in the small progress bar, so use the Undo Graph trick to monitor your progress.
Local Wiggle: Keep slicing and dicing; it never seems to stop working, though it does slow down a lot towards the very end, and will start returning negative scores with almost every combination you try. I usually stick with pairs of segments, but occasionally a large (like, 1/2 the whole structure!) will work too.
Pulls: If you lock off sections as you do for the Local Wiggle, but instead do a pull on those segments, you'll often see a gain in score. Watch the undo graph to see if your move is good; if the score drops move quickly in the other direction instead. If you do it too long, you've moved many pieces out of alignment and won't get them back. Just Restore-Best and try again with a different starting direction. One or two attempts per "locked section" is all that is needed.
My best results come from locking off section 3-4 pieces wide in an "all-loop" structure and move the middle one or two pieces up and down. Also look at helix-like sections in that all-loop structure (but you probably don't want them to actually *be* helices, because those are much stiffer and move less) and pull them "left and right", that is, you're trying to move them closer or further to their neighbor coils. This seems to act like a very fine-tuned version of the "Small Rotate" command above.
Sidechains : Walk the protein's sidechains, adjusting one to at least one new position and do a wiggle to test it, and a nudge-wiggle too. If it didn't help your score, restore best score and move on to the next sidechain and try again. Don't forget to re-shake after a score gain (or even a score drop!) - the game may find a new position for another sidechain and help you out.
Rebuild The Ends: I'm pretty bad with the rebuild tool, but sometimes have luck rebuilding just the last two segments at each end. I let it runs for a few positions, pick the highest one and do a local wiggle on those last two segments.
These are some rules of thumb that I usually follow throughout the game.
After any gain in score I do a shake (global and all the local shakes) to "lock in" the new structure. I'll probably try a global wiggle too, especially if the shakes produced any movement, to see if we've just made through a wall and the structure is now ready to roll. "Ready to roll" means its going to start yielding points and just keep going and going, far beyond the tiny gains you'll see most of the time. A roll is cool thing to see happen :)
I usually play with the structure changed to all loops, but I create helices so the Tweak tools will have something to work on. If I'm really stuck I might try putting a more conventional set of structures in place (i.e create all the sheets,helices etc as I think they should be) to see if some of the tools will work better on them.
These days I pay almost no attention to hydrogen bonds: they seem to take care of themselves and I don't have them visible. This is an interesting contrast to when I first started playing: most conversations were about how many h-bonds we had between which sheets. MOST of the time they will naturally fall into the right places, but watch out on "barrel" proteins, or other proteins with lots of sheets. In such puzzles the sheet positions are very important and worth experimenting with.
I use the Undo Graph to monitor my score almost all the time, except when making big moves that change the score by large amounts.
Constraints in a puzzle will tend to go away on their own, so if I can't get rid of them with a minimum of tweaking or a small rebuild, I usually don't worry about it and figure they'll eventually go away when the structure naturally gets to a new layout. Or I wait for my team-mates to do it for me :)
What I Don't Know
Rebuild is still pretty random for me, so you'll need to read other people's ideas on those.