Road marches, and the relevant field manual, are primarily concerned with administrative, i.e. non-tactical, movement from point to point. Generally, it's for getting a group from here to there when bad things aren't likely to happen.
Moving when there is that risk is properly known as patrolling, and addressed better in the text on that subject: FM 21-75 Soldier Combat Skills (January 2008). It was formerly known as Combat Training Of The Individual Soldier and Patrolling.
The part of the current manual that concerns us is chapter 7 (p. 123 of the pdf), Individual Movement.
The primary individual movement is walking.
The difference between walking on a march, and patrolling, it that on a march, getting there is the whole point. On a patrol, getting there is half the fun.
Before you gear up, everyone on the patrol should know the entire patrol cold, from start to finish: Where you are, where you're going, how to get there, what route, all actions in case of a myriad of contingencies, and what you're headed out to do. There should be a designated chain of command from leader to the very last man regarding who's in charge if something happens to the person(s) above them. Once you get outside your own wire, there's no guarantee who's going to be running the show when the patrol when they come back, and bad times are no time to decide, let alone argue over, who's in charge. And everyone should know what to do if they get lost, separated, or have to make it to a rendezvous or return to their start point alone and unassisted. (We'll cover more of this, but understand the general concepts overall at this point.)
Before you start out, even more critically than on a simple march, you should be inspected by the patrol leader to ensure you've got everything you're supposed to have, and nothing you're not supposed to have. Radios should be checked, weapons (if applicable) should be function-checked (and test fired if practical/possible). Nothing on you should rattle, jingle, or shine. You should be able to test-walk several yards and not make any sound above your actual footfalls. Everything should be properly stowed and secured. You should be wearing eye protection, and if you have it, hearing protection that will allow normal sound, but block damaging sounds.
Weapons are carried locked and loaded once you're out the gate, not slung on your shoulder. On a road or trail, half of them should be pointing to each flank. The person carrying them should be using three senses of five at all times on a patrol: looking, listening, and smelling. Anything and everything you collect with those senses may be directly significant to either your life expectancy for the next minute, or for information-gathering purposes, to build an intelligence estimate of your area under whatever the current circumstances are.
Whenever someone is on a patrol they should be looking at their assigned area, with the muzzle in the same general direction. As well they should be regularly doing quick scans of the persons ahead of them, the persons behind, and/or to both sides when moving in any formation or terrain. This is so they can see a change from anyone rapidly, without the need for other communication, and note indications to halt, freeze, or take other action, depending upon what is going on. That way, on a patrol, multiple sets of eyes look over everything. Along the entire route.
You should be listening, for anything that doesn't fit, including nothing. The world, even in nature, is noisy. Dead silence is usually a noteworthy sign. So is everything you hear.
Smell is equally important. Smoke, exhaust, cooking fires, food, dead animals, etc. all activate a primitive part of your brain, for good reason. Your lizard brain knows things your monkey brain didn't get taught in school. It may sense something that makes the hairs on your skin stand up. you may smell something or someone, before you can see them or hear them; pay attention, and pass it on when it happens.
There are multiple reasons to patrol.
It may be a reconnaissance patrol, to see what's out there.
It may be a combat patrol, to find bad people and do bad things to them, or wait in ambush for them to do so.
It may be a logistical patrol, to get or take supplies from one place to another, or to deliver or collect people similarly.
Because of this, as a rule, a minimum patrol should probably be three people. If one gets injured, the other two can bring the injured back. One can keep watch, while one sleeps, and one communicates/relaxes/rests/eats/poops/etc. You can do a patrol with more, even many more. But generally, less than three people will make things a very difficult time if anything unexpected happens, and patrolling is where the unexpected is the reason you're patrolling in the first place.
Besides walking, you may need to move through areas more surreptitiously, either because of lack of concealing vegetation or terrain cover, or the possibility of getting seen/shot/killed.
Your alternatives at that point are approaching stealthily, rushing, high crawling, and low crawling.
Stealthy simply means slow, steady, deliberate movement, avoiding making any sound or sudden movements, and taking full advantage of any concealment, shadow, and cover in the area. Generally, you're still walking, whether upright or crouched.
Rushes are for open terrain, where speed moving from one piece of cover to another is more important than the risk of visibility by rapid moving, because you either figure you've been seen already, or the incoming fire makes that obvious.
You use rushes to move up on someone who knows (or should) you're coming, but dropping back to a covered spot as rapidly as possible, before you can be sighted and shot. Alternating pairs can approach a position with rushes that prevent someone from concentrating fire on a single person or pair.
High crawl is on your knees and elbows, belly and gear off the ground, head up. It allows someone to cover a lot of ground either quietly, or at least quickly, if the brush or cover is only a couple of feet tall, without them being seen. If you're relatively young and fit, you can get across 50 yards in a minute or less.
Low crawling is flat on the ground as low as you can get, generally because cover is extremely sparse, and/or incoming fires are sweeping the area being crossed.
What happens when you stick up too far?
Skip to 2:33 through 2:45:
There's a lot more to cover; this is just the start.