Step #0: Encumbrance Slots
Your character can only lug so much around. By default, each character has six equipment slots (head, body, left and right hands, legs, and feet). A character can fit an item in any of their slots up to the limits of their Strength modifier +1. So if you have a Strength of 8, you can put anything with an Encumbrance of 1 into any of your slots. You aren't limited as to how many of these slots you have filled: a character with a Strength of 8 can fill all six slots and they'd be fine, as long as no individual slot had an item in them that was more than Encumbrance 1.
This is an abstraction for the purposes of simplicity, so if you want to get more detailed (measuring exact poundage and location of every bit and fob), you can. It might bog down play and make the thing more into an accounting exercise, however.
By and large, there's three major encumbrance categories: light (encumbrance 0), medium (encumbrance 1), and heavy (encumbrance 2).
Step #1: Supplies
When you're preparing to go camping in the woods, it pays to do some forethought. Namely, in the form of supplies. Again, abstracting for the purposes of simplicity, one days' supplies cost 5 gp. This includes food and drink, but it also includes tents, bedrolls, appropriate clothing, tinder, healing herbs, firewood, repairs to your existing equipment (re-forging a knicked sword or repairing a holey shoe) to prepare it for the wilds. You can buy more than one days' supplies if you want, for an additional 5 gp per day (you spend it on better equipment, better rations, etc.).
Now, supplies are heavy things. One day's supply is Encumbrance 1, and it goes up by +1 Encumbrance for every additional day you stack onto it (that is, two days is Encumbrance 2, three days is Encumbrance 3, etc.). An average STR 10 person can't carry that much on their own shoulders.
And this is why people domesticated pack animals and invented carts. Animals are pretty hardy, and can often carry several days' worth of supplies. Carts and wagons are even better for long journeys, and can carry weeks or months of supplies. They can cost a little extra, but provided they survive the journey, they can be a good investment.
Step #2: Daily Travel
When you set out for the day, your party travels at the speed of the slowest individual, and marches for about 8 hours straight, with a light lunch in the middle. If nothing gets in your way, this marching is fairly simple, and you can cover about 24 miles over flat, good terrain (such as a paved path).
Of course, things can get in your way. The DM can roll at any point for a "random encounter" along the way. They may do this secretly, before the game even starts, or they might plant encounters along certain routes, or they might roll on the table in front of you. They might do it once, they might do it over and over again, they might do it more in response to actions your party takes or areas your party travels in. These encounters might be with creatures, or they might be interesting scenes, or unusual events. Creatures encountered might be helpful, harmful, or neutral. The possibilities are pretty endless, but you can bet that the more wild and untamed the land, the more dangerous and monster-riddled those encounters are likely to be. When you're traveling close to town, you might meet a pilgrim or a town guard or a stray dog — hardly real threats. Deeper into the wilderness, you might encounter obscured paths that might risk you getting lost, or bridges that have been washed away by floods, or violent wild animals, or bandits along the path. In the deepest wilderness curious fey creatures and horrible monsters may lurk. You might get hit with a rainstorm or a snowstorm wherever you are, if the season is right. Sometimes, your cart just breaks, or your well-trained horse decides to bolt from some unseen threat: accidents do happen, even in the safety of town walls.
If nothing happens, you use one days' supplies at the end of the night when you take an extended rest (you repair your shoes, you eat your meals, you start a fire, you make camp). Your load a little lighter, you start again in the morning.
If anything happens, things could turn ugly.
Step #3: Random Encounters
One thing remains true as you're out exploring the wilderness: the longer you stay out there, the more likely you are to meet something, and the more dangerous the area you're in, the more likely that something is going to be a problem for you.
DMs have various methods of whipping up these encounters, but regardless of their method, these facts remain true.
When you do get in an random encounter, your DM will introduce it to you, and then you can play out the encounter like any other scene. For instance, if the DM determines that you come to a bridge guarded by trolls, you can decide what to do. Do you bribe the trolls and hope to get through? Do you fight them to their bloody end? Do you try to find another way around?
You resolve the scene using ability checks, and your character's specific class abilities. A rogue, for instance, might be able to sneak past those trolls and then distract them long enough for the rest of the party to cross. A wizard might [I]charm[/I] them into letting everybody pass.
Typically speaking, a random encounter will require one successful check for each PC present to get past, though there are certainly easier and more difficult encounters in the mix, and not every "random" encounter is truly very random.
Random encounters might also not be very big scenes in the typical encounter sense. A rainstorm might be a random encounter, and while you might describe how your character reacts, there's little to do to "fight against it." A broken wagon-wheel might be a random encounter and, similarly, you might want to make a check or two to fix it, but you might also just load up the horses a bit more and leave the wagon behind. A random encounter might be some moss on a tree that kind of looks like its a face. Maybe it's a treant, maybe it's a red herring, maybe either way you give it a wide berth. These encounters have mechanical effects, but they don't involve a lot of interaction or detailed obstacle-overcoming. How you react to these challenges is up to you and, depending on your reaction, the DM will apply the results.
Step #4: 1d100-1 Problems (reroll bitches)
Random encounters encompass a broad swath of potential things, but ultimately they affect your character in a few concrete ways: your party can either gain resources, lose resources, or have their resources unchanged.
By "resources" here we mean, mostly, supplies, and whether or not you're suffering from some ongoing problem, termed a Complication.
The first case, supplies, is pretty straightforward. If a goblin thief runs off with your horse, you've lost all the supply days you've put on that horse. If a sudden rainstorm mires your wagon in mud, you may have just lost that wagon and anything in it (unless you can make ability checks or use class abilities to un-stick it).
The second case, Complications, covers a rather broad swath of problems that all share a similar solution: they go away only when you make successful ability checks, use a special ability or item, or when you return to town to pay to make them go away.
Complications are usually applied when you try to accomplish something in an an encounter, and fail. Perhaps you get knocked out in your fight with the trolls, perhaps when you tried to get the wagon out, you weren't strong enough, perhaps you tripped the trap, perhaps you didn't notice the pothole, perhaps you failed to identify that adorable bunny rabbit as being a voracious beast that will consume your soul…you know, normal fantasy world troubles.
Complications may take the form of diseases, poisons, curses, long-term injuries, penalties on your speed (which ultimately cost you extra supplies, AND mean you might get another encounter!), continual supply loss, or any number of other effects. What all complications share is that they do not go away when you rest — they only go away with successful checks or the use of certain abilities or items. They also may get worse or better over time. A disease might wind up seriously risking your life, while a wound might get infected, or that soul-consuming thing might spread to other party members. Not cool.
The difficulties complications impose might hurt your ability to make ability checks, might restrict your ability to use certain items or class powers, might hurt you in combat, might cause you to consume more supplies, or, perhaps most painfully, might hurt your party's speed (which means you spend more nights in the wilderness). They might also reduce your maximum HP, which is a bit of a problem, not only making you weaker in combat, but also potentially killing you. If your maximum HP is reduced to 0, it is exactly like you have been brought to 0 hp, only worse since magical healing and recovery do you no dang good. If a complication brings you to 0 hp (or less!), the only way out of the situation is to remove or lessen that complication!
Running out of supplies introduces its own complications, including exposure, starvation, and thirst, each of which might inexorably murder you just as pitilessly as a goblin, but probably much more slowly. If you run out of supply days, you have no food, no water, no repair materials, no medicines, no firewood, dilapidated tents, flea-ridden bedrolls, and probably a hole in your shoe. Good luck hunting.
For those of you who can't follow the logic, it goes like this: Spend time in the wilderness, get in troublesome encounters, lose resources, get a Complication, starve and die bleeding and alone with only your soul-devouring rabbit for company. No amount of rest will save you. Clearly, spending time in the great outdoors in D&D is a risky proposition at best.
Step #5: Rest and Recouperation
At the end of each day of travel, the party can rest. The DM, however, might still roll for an encounter, if they want. Even when your party sleeps, they aren't necessarily safe, and some monsters or critters [I]love[/I] attacking you when you're at your most helpless. If you're safe in an inn room in town, you're likely OK, but even there you have thieves and spies and all sorts of potential problems for your restful night's sleep.
When you take an extended rest at the end of the day, spend 1 day of supplies, representing the food you ate and the repairs you made and the fires you burnt. If you can't do this, you (a) can't take an extended rest, and (b) start to risk Complications resulting from starvation, thirst, and exposure.
Generally, if you're staying at an establishment, you won't need to spend your own supplies, and can use the inn's food, clothing, etc. Sometimes for cheaper! Sometimes not.
When you take an extended rest, you can also make use of abilities and ability checks and items that might help with any of your complications. Use them to see if you still suffer from your complications after this rest. Keep in mind when you restore your HP and HD that any complications you suffer from that reduce your maximum HP are accounted for. And then get back out there!