Zig-zagging across a room is equally as fast as running straight. (Horizontal movement has no effect whatsoever on vertical movement, or vice-versa).
Hugging Walls
Run diagonally against SMOOTH walls in the direction you intend to turn next, so that you'll "turn the corner" on the 1st possible frame and not lose any ground.

Run diag on Stairs (slopes)
If you run straight at a slope, you will advance forward in an alternating pattern of 2-0-2-0 pixels every frame, while also getting pushed 1 pixel aside. So this is the same speed as if you were just walking.
Even & Odd pixel
While Dash is held, you move 2 pixels per frame on each axis you're moving. This means your coord stays Odd or Even as long as you don't hit obstacles.
Changing Even/Odd
Odd Walk left against a Left wall, or clip up/down a Left wall's corner)
Even (also room default) Walk right against a Right wall, or clip up/down a Right wall's corner)

"Clipping" corners/slopes
Clipping in CT refers to running straight into an obstacle and getting pushed sideways. The result is a 0-pixel distance between you and the side of the object.
Bird Skip (Truce) The one skip where Y-coord matters. Hold Dash before coming off the ladder, and stay to the very upper side of the bridge (Odd Y). Once you're on a good X for the skip, hold Dash BEFORE moving up (to stay on Odd Y).
Waterfall Skip (Denadoro) Your coord MUST be even, so hold Dash before the room loads and don't bump into any corners. To ensure success, WALK against the Right wall to fix your X. (Y coord doesn't matter, just don't start too low and totally miss the clip)
1st Dactyl Skip Pixel-perfect and lacks good visuals. But the specific coord you need is even, so by keeping Dash held (moving always 2px at a time), you can pick out the good position more clearly
Bat Skip (Heckran Cave) Clip straight Down (for just a bit)

All hitboxes for sprites are a maximum size of 1 tile (16x16 pixels), even for sprites that appear quite large. Most fights and traps are activated by touching either an enemy sprite, or an invisible sprite ("hot tile") on the field.
When you stand still, your hitbox is smaller, and corresponds roughly to the visible space you're standing on. But when you move, your hitbox extends out fully in the direction(s) you press. Try approaching a friendly NPC head on, and notice when you collide that you've stopped several pixels from it. Now walk past the NPC instead, trying to brush past its side; you'll see that you can close that gap entirely without colliding. This applies to enemies and hot tiles as well — you can squeeze past safely, so long as you don't "look them in the eye", or cross their path of movement.

Room Triggers (how to skip them)
Exactly upon room load, event triggers begin cycling in intervals of 28 frames (the time it takes to Dash across 3.5 tiles). Depending on the 1-in-28 that you touch a trigger, the game takes away player control a certain number of frames after (ranging from immediate, to like 12 frames / 1.5 tiles later). Therefore the trick is to cross the trigger at precisely the point in the cycle that affords you the extra time to run 1.5 tiles (where you can save your game). The setups for these absolutely REQUIRE that you begin running forward on the 1st frame possible (i.e. hold Dash+Dpad BEFORE the room loads!!!!!!!)

TAPS ("Release-B" / Release-Dash)
Sometimes also a number of "taps" are called for (release Dash for the shortest duration you can muster: ideally 2-3 frames each). These taps must be performed WHILE moving (releasing Dash at all before moving = not good; you need 2-3 frames of walking speed).

Attempt to menu on the tile past the trigger. If you menu early, just continue running and mashing menu. Both you and the trigger cease/resume movement when you enter/exit a menu, so there's no problem. In fact purposely menuing early is a safe practice.

Yakra Skip actually an uncommonly lenient trigger (because you can just save on top of it, rather than having to reach the other side). Still, enter the room running.
Heckran Skip running straight Up into the room, 1-Tap while running. You can actually make a safety save when you first enter the room, but note: be mashing menu WITH your Run buffered, so that even if you menu a couple frames slow you may still attempt a proper skip without reloading.
Captain Skip running Left, 3-Tap while running (gotta also jog a bit up then down to avoid the Captain). If performed correctly, the game will let you run a full tile past the fallen guard's head! So just practice the movement to see if you're accomplishing this, before even adding in the menu. Once you have the movement down, try to menu with Crono's foot just a few pixels past the fallen guard's helmet.
Zombor Skip running Left+Down, Tap 1x while running. Try to get shoulder-to-shoulder with the guard (or within a couple pixels behind him) when you menu

"Regrouping" / Whipping into formation
Try to get party members closer to where they will walk in cutscenes. Examples: discovering Robo (get Marle close to him before checking), getting Broken Hilt from Frog (prevent Lucca from moonwalking across the room)
Approaching specific fights (like in Magus' Castle), run Right-Left-Right to "whip" characters into the triangle formation beforehand.
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