The main thing to remember here is that Halo 3 is very forgiving, compared to CE and 2. Enemies will not be as aggressive as they were in Halo 2. That doesn't mean you should take it easy.
This won't be a step by step guide. More so, I will explain a lot of handy tricks that will decrease your time, and also advise you on which levels to optimize, and which ones you can be lenient on. Doing it co-op will be easier, but it is not imperative.
My times for Halo 3 Legendary:Sierra 117 - 18:08 (CO-OP)
Crows Nest – 16:18 (CO-OP)
Tsavo Highway – 12:51 (CO-OP)
The Storm – 13:40 (CO-OP)
Floodgate – 4:22 (SOLO)
The Ark – 20:57 (CO-OP)
The Covenant – 28:24 (CO-OP)
Cortana – 8:13 (SOLO)
Halo – 12:42 (SOLO)
Total: 2 hours 13 minutes 35 seconds
As you can see, I have over 45 minutes to spare. That's huge, considering every minute saved is super important.
Levels you should optimize:
Floodgate
Recommended time: 5 minutes or lessThis is without a doubt the easiest level in Halo 3. You should aim to beat this in 5 minutes or under. There are no tricks that you need to learn, but knowledge of the level is required.
Cortana
Recommended time: 15 minutes or lessOkay, I know what you're thinking. One of the most notorious levels in all of Halo. It is actually exceptionally linear and relatively simple to speedrun.
Halo
Recommended time: 16 minutes or lessThere's an awesome trick you can do at the beginning of the level which allows you the skip the entire ascent up the spire. I highly recommend you learn and practice this, as it will take 5 or so minutes off your time. My solo speed run is below. Timestamp for the skip is 55 seconds in. If you're doing it co-op, the person playing as Chief will need to do the skip, as frag grenades are essential. You need the grav hammer from the first enemy you encounter.
Crow's Nest
Recommended time: 22 minutes or lessThis level has one difficult section, which is the hangar defense. Therefore playing co-op for this level is highly recommended. During the hangar section, make sure to pick up a deployable cover, as you can use this for a neat trick later on. This level has a number of neat tricks which are relatively easy to pull off. For example:
^ You need the grav hammer for this skip. It looks hard but it is very easy to pull off after a little practice. The host must perform this due to latency.
Drone section made easy (deployable cover needed, can find during hangar defense).
Tsavo Highway
Recommended time: 16 minutes or lessCo-op is highly recommended to speed through this level.
Other Missions:These missions should be attempted after the above ones.
Sierra 117
Recommended time: 25 minutes or lessIn my opinion this is actually quite a difficult level to speedrun. Very close quarters with little cover makes it less desirable for speedrunners. 3+ players is highly recommended here.
The Storm
Recommended time: 20 minutes or lessThis one isn't too bad, just longer and tedious. There's an awesome trick you can do for making the scarab fight super quick. Basically you use a grav lift to jump on the scarab early. The video below will show you exactly where it is and how to do it.
The Ark
Recommended time: 25 minutes or lessWith 2+ players, this mission is a breeze. It's just long and arduous. Tank beats everything.
The Covenant
Recommended time: 35 minutes or lessVery long, but easier with 3+ players. There's a number of neat little tricks you can do on this level. One being at the start of the level.
During the scarab battle, take out the banshees first, then focus on the scarabs. Once a scarab is temporarily disabled, you can either jump onto it and take it out manually, or shoot off the back with the hornet and take out the core that way. The hornets are very vulnerable so you need to strafe to avoid the scarabs laser.
I hope this helps. Feel free to comment if you have any questions.