I know it's only going to be marginally useful, but there are only best practices that I can provide rather than concrete answers to so open-ended a question. Even if you had specified which levels, should the levels in question either take shift control out of your hands or lack shifting altogether, it all comes down to your FPS platformer navigation skills.
- There is no "run". You can only walk at one speed really, but the use of strafing can possibly help speed up your movement - if forward movement and sideways movement are equal, diagonal movement, using them together at a diagonal would be faster (think of a square representing movement speed), but I haven't confirmed whether this is the case, even though it would appear true from my own goal time runs. Jumping seems to be slower than walking so jump only when necessary.
- Speech progresses on its own. It is usually triggered by passing through an invisible trigger volume in the level or performing some specific trigger interaction and will be skipped when another speech segment commences. To speed up segments dependent upon waiting for some speech, quickly reach the triggers to begin the speech as early as possible.
- Learn the area and map out the shortest route, cutting every corner as short as possible and jumping as little as possible. Remember that a straight line is the shortest distance, but also avoid clipping on furniture or corners as that can ruin a high speed run immediately. Always look for shortcuts to avoid whole areas or certain actions and believe me when I say that there are many shortcuts, but that some of them require a high degree of precision.
- Once you are confident you've learned the level and are taking the correct approach, quickly restart the level as soon as you screw up any given attempt to avoid wasting time on a failed attempt. You will be doing a lot of this.
- When you can't control the shifting, learn Desmond's timing and take risks, rushing to make use of every shift to the fullest of your ability for the duration of the shift. Some of these can be infuriatingly tight, but are necessary for some of the goal times.
- While sometimes fewest shifts can be synonymous with goal time, often they tend to be objectives in opposition to one another. Do not be afraid to shift as much as necessary to get the goal time and worry about shift goals for another run.
- There are several stock shift maneuvers that the game has you perform that you should learn and learn well. Quickly fluffy+slow throwing projectiles for you to ride or catch elsewhere in the level; Upside-down/normal surfing/catapult/projectile levitation, especially timing to maintain momentum; Quick fluffy+heavy throwing projectiles through glass; Upside-down+slowing objects for you to hop across; Shifting mid-jump to let things fall or raise as needed.
- Don't be afraid to skip sections even if the speech tells you you will need to do it. It may be harder without going through a certain section here or there, but it will invariably save you time.
If you have questions about specific levels, feel free to create questions for them and I will answer as best I can as I've gotten the goal times and shift goals on every level.