I've created this thread so it can be used to discuss all Tomb Raider games (1-5). I'm deeply familar with the first installment and much less so from TR2 onwards. The engine and exploits are largely generic.
All the below information relates mostly to the first installment.
TR1 is much more puzzle and trick based, and the stages are generally a lot smaller. The current WR for the game is
1:19:27 by RadxxRyan. The best goal for the game would be to use ingame timer which is displayed at the end of each level. Pausing does not count towards the time, but to compensate the real time argument picking up items costs time and I know that in TR2 pausing the game will still cause the ingame timer to increment. This is why the PC version is favoured since healing and equipping different weapons can be macroed.
Walking
The short delay after pressing walk can be exploited to your advantage. If you run towards an edge and hold walk as you get close you can immediately stop at the edge without any walking at all. There is also another useful exploit called the walk-run jump where you take one step forward it will allow you to do a quick run jump in a very small area
Running
Standard movement this should preferably be avoided whenever possible.
Standing Jumps
Faster than running but should still be minimized in favour of running jumps. This movement still has a place because if you let go of the D-pad before Lara lands she gets a small stun animation and can imeadiatley jump again. This is useful in areas which have condensed platforms.
Backflips
Might as well mention this, they get you higher and further than a standing jump and also give Lara a smaller hitbox. Not sure if this will have any use. However, it will be used a lot in other TRs because of the availabilty of a mid-air roll.
Running Jumps
The fastest (horizontal) movement in the game. Basically you need to try and fit as many of these in as possible. Levels may require extensive route planning to maximize it. In areas with high or uneven ceilings you want to try and jump in the area with the highest ceiling so your jumps are as long as possible.
Moveable Blocks
Preferably if possible you should try and push them as it's marginally faster.
Pendulum Jump or Swing Jump
Sometimes when you jump you land part way into a block and need to immediately do a running jump from there. If you just run forward you will fall off, however, if you "swing" slightly to the side (so you cover a little more ground) before turning forward again then you can usually cover enough distance to start a running jump. The discovery of the walk-run jump has mostly obsoleted this technique, but they can be combined for the shortest distance running jump setup possible
Avoiding Bashes
If you hit a wall in the middle of a jump then you bash against it and get knocked back, however, if you do a grab in mid-air you do not. This has all manner of uses.
Safety Drop - Grab/Bug
If you fall backwards off a ledge and press grab WITHOUT holding it you can land on ledges below you (if you hold it you will just grab the ledge again). In some cases this can save a lot of time. This is sometimes labeled a "bug" due to it apparently not being intended by the designers.
Swing Cancel
On very thin ledges Lara gets an increased swing animation which causes her to take an extra couple of seconds to pull herself up. However, if you hold action let go of it and re-apply it late on you can avoid it.
Curve Jump
Possibly the most useful advanced technique of all. Core seemed to have not taken into account the curve jump possibilities that various levels have, allowing you to break huge sections of them. Curve jumps come in several forms; some are almost effortless, while others are incredibly hard.
Cancels
Cancels refer to either trying to avoid stumbles altogether or avoid a large one since a small stumble can be cancelled with a forward jump.
Trip Cancel
If you roll near a ledge Lara twitches then falls off, thus rolling out of a stumble.
Grab Cancel
This seems to be a technique exclusive to Tomb Raider 1. If you are making a jump where you land in a large stumble but DO NOT take damage from it, then grab just before Lara puts her arms in the air (the point where grabbing becomes impossible), and you will land in a small stumble instead.
Action Cancel
When falling from a distance which would normally cause a stumble you can hold the action buton and pickup an item or use the items in a keyhole or pull a lever.
Ceiling Bug
If Lara touches a ceiling before the peak of her jump and lands on the ground. Provided it does not cause her damage, then the stumble will be non existent and she can continue running normally.
Corner Bugs
This is very useful technique throughout the game. A corner bug will cause Lara to zip vertically until she lands on a ledge or hits a ceiling. It generally happens when she passes through a corner at 45 degrees. There is several variations of it however.
Standard Corner Bug
Most commonly performed on a console and should be avoided when TASing. When doing standing and jumping Lara moves a few inches forward, this effect will cause Lara to be imbedded in a corner.
Arch Bug
Arch bugs exist where the oposite side of the corner is covered by a ceiling. Lara needs to jump once to imbed herself in a corner then she can run forward and appear on top of the surface.
Water Corner Bug
There is two methods to do this. First one involves Lara swimming of the surface. If she lines herself up much like a standard corner bug and then holds the swim button, she can imbed herself in the corner and then get zipped up. The other version involves swimming below the surface, getting imbedded in the corner and then turning to bug Lara up.
Trigger Bug
Since the game consists solely of squares. Many events are triggered by going over a square such as enemies, room states and predetermined events. Some can be avoided by jumping though a corner thus avoiding going over a specific square which is attached to certain trigger code.
Collapsible Tile Bug
If you sidestep on a collapsible tile then Lara will be warped to the ground unharmed.
Ground Warp Bug
If you hold the walk button while being on edge of a ledge and an agressive enemy attacks Lara then she'll be warped to the ground unharmed.
Crack Bug
For corners which are facing inwards, its sometimes possible to pass through them. This applies most to moving blocks
Flying Corner Bug
This will be used extensively throughout a run. You need to jump through a corner at 45 degrees and then turn when passing through the corner to bug up the wall or platform. The following video best explains it.
http://uk.youtube.com/watch?v=ju68MrndT18
Fence Bug
If you run at the seams its possible to pass through them as if they weren't there.
Dive Bug
A very cool trick but requires a lot of precision. If you dive from a height which would normally be fatal and hit the ground and a wall at the same time. then Lara will be virtually unaffected. The following video best explains it.
http://uk.youtube.com/watch?v=TgKRNOyOdok
Wall Slope Bugs
Wall slope bugs seem exclusive to TR1 mostly because Lara's stun animation was changed in the consequent games. Quite simply if there is an incline into a wall then the chances are Lara can run into it, and jump at a precise moment and then bug up the wall. For this to happen Lara's feet need to be partially glitched into the ground while colliding into a wall. The following video best demostrates it.
http://files.filefront.com/wall+bugsavi/;7601900;/fileinfo.html
I hope thats mostly everything. If I've missed anything I'll add them later.
