I just beat the campaign on Legendary solo today (only played a mission or two a day for a week or so) having previously beaten it on Heroic. In many ways Heroic prepped me for Legendary. Many times I found I had alot less trouble with the same parts on Legendary than Heroic because I knew what to expect.
Tricks I used to in Legendary were as follows:
Charged plasma pistol to take down elite shields quickly.
Needler works great against brutes who have no shields. Pump off half a clip and run and the super combine explosion does the rest. Further, many grunts have Needlers so ammo is generally not an issue.
Anything that has a scope for sniping is good. The Magnum works well against grunts and you wont have to waste your precious DMR ammo on them. In general keep at range when ever possible and make sure every shot counts (headshots galore).
Whenever you see purple shields, there is an engineer (floating blob like thing) nearby. Always kill it first, but be extremely careful to not expose yourself while doing so.
Get check points whenever you can but be careful to not get them if you wasted a bunch of ammo from trying to kill one enemy. I found that it's often better to die and restart at a checkpoint than to continue with low ammo. And there are alot of places where check points either spawn to frequently for their own good or not frequently enough. Always be aware of when and where a check point is triggered.
Avoid the super brutes (not sure the name of the big green heavily armored ones who take forever to kill) whenever possible. In many parts, you can simply go around them to complete a mission objective. Just watch out for their weapons.
Additionally, avoid combat whenever possible. There are parts where the cloaking device is useful to simply get around things. Save that ammo and health for when you really need it.
Spoilers: On the almost final part of the last mission, the 4 elites, couple of grunts and an engineer, the hologram worked wonders. I was able to take out the engineer while the elites were chasing my hologram and run back out for the check point. I then proceeded to take out the grunts quickly with the magnum (found in the storage shed in the previous area). Once they were dead, I used the Hammer on the elites (which worked surprisingly well). After every kill I ran back out into the previous area to get a check point.
Further, on the very last part, make sure you have full health before you trigger the check point near the giant cannon. I had minimal health, so whenever I'd die, I had to run back to get a health pack which was extremely annoying.
One final mission area I had alot of trouble with was the 6th mission after the sabre fights. You land on the covenant corvette and need to enter through the giant shield. Ultimately, I found that, once you kill the 2 jump jet elites and your allies drop down to their deaths, you are left with like 4 or 5 elites to contend with and no check points in sight. That's not entirely true. On a whim I decided to explore around the top of the corvette for a while and eventually the next stage mission sequence triggered. I was then able to get several check points which helped immensely. Then it's just a matter of dropping down to the top most catwalk (dont drop all the way down until all elites are dead), fire alot of rounds at an elite while avoiding fire yourself, jumping back out of the shield, letting your shield regen, checking for the next elite target you are going to take down, and repeat. This part is a testament to your halo skills. I had more trouble with this one part than any other in the whole game (except maybe the "endless wave" part on the final mission when I first played it on heroic... Second time through was actually really easy).
Youtube has alot of decent videos on getting through the game on legendary. I used them when I got completely stuck in parts and they are really useful. Just search halo reach legendary campaign and a bunch will pop up. Probably add the mission name as well.
Anyway, those are my experiences with Legendary solo. I hope they help others with getting through as well.