You're going to be doing a lot of this. Like Morrowind, the dungeons (caves, mines, ruins, whatever) are very straightforward, mostly linear and fairly difficult to get lost in (which is why you're not going to see a lot of maps here). So learn to love your map (F4, by default in the PC version). Unless you've got a constant effect item with Night Eye, seeing everything in most dungeons is going to be very difficult, so your map will show you some of the things that you might have otherwise overlooked (like doors to other areas, for example), but will not show you things that should not be overlooked (like traps).
On a positive note, the devs were very kind to players in their dungeon designs. In many cases (and this is particularly true in Ayleid Ruins and Forts) you will not have to tromp back through the entire dungeon to get to the exit. For the most part, there is a door or portcullis somewhere near the entrance that cannot be opened from that side or that you can't reach, but it can be opened from the other. Caves will sometimes have a hole from an upper area where you can drop down into the entrance area to get out. But since there is no Levitate effect, you can't take the short route to get to your target. So, in a lot of cases, once you work your way through the dungeon, continuing to move forward will eventually put you somewhere near the entrance.
At various points in your career as an adventurer, you are going to need to rest. The only time that this is required is when it's time to go up a level or if you want to heal up quickly. For those, 1 hour is all you need. Anything else is just a role-playing thing or killing time. Resting may be somewhat problematic in a dungeon setting. Here's a good rule of thumb: only NPCs need to sleep in a bed. Undead and critters (daedra included) don't. So if you enter a place and run into humans, there will be beds around somewhere. If all you are seeing is critters, vampires, undead, and/or daedra, there probably isn't any place to rest and you'll have to leave the dungeon to find one. So either check how far you are from advancement before you enter, or wait until you leave to find a resting place. With very few exceptions, dead creatures will not respawn while you're still in a dungeon, so seeing the "enemies nearby" message just means that you have to clear out the area and after that, it should be safe to rest for as long as you'd like.
Some equipment should be standard. A weapon of some sort is almost required unless you're a pure spell-slinger or role-playing a monk. But I'd recommend two: a melee weapon matched to your skills and a bow. I cannot overemphasize the importance of being able to get in at least one really good sneak attack before your enemy even knows that you're there. At the very least, a missile weapon will allow you to knock down those Welkynd Stones and Varla Stones that the devs like to place so high in Ayleid Ruins.
Spell-slingers with high enough skills in Destruction could accomplish much the same thing without being able to get the extra damage bonus for the sneak attack. But on the other hand, if you detect them far enough away, you can fry them while they're still charging. I've done it plenty of times in the Arena. It works great and even if they're still breathing when they actually reach you, they've been softened up enough to where weak combat skills on your part shouldn't present a problem.
On the subject of spells, there are a few that you shouldn't leave home without. You start the game with a Heal Minor Wounds spell. It won't eat up much magica, but it's only a workable solution if there are no enemies beating on you or you have time to cast it fifty-bazillion times at higher levels. So invest in better spells, or tote Healing potions. You'll come across lots of potions or make your own, so those are not an issue. Scrolls can be a bit more problematic. They aren't as common as potions, but they are out there (enemy spell-casters usually carry a couple and some Dremora are almost walking magic shops). Very seldom are they healing spells, though.
In addition to Healing, make sure that you have some way of restoring damaged attributes. Several creatures use effects that damage your attributes (not drain or absorb - those wear off). Either invest in restoration spells or buy/make potions. By the same token, carry around at least a couple of Dispel potions. Spells are all fine, well and good unless you've been silenced and can't cast. Even though none of the Silence effects lasts for more than a few seconds, that's plenty of time for someone to open a big ol' can of whupass on you before you can throw up your defenses.
I take it as axiomatic that "knowledge is power." So the more you know about your enemy (like where they are), the better off you will be. Regardless of your class, get something with a "Detect Life" effect. It's a very low-level spell that almost anyone will be able to cast. Even in the dark, you will be able to locate any living thing (including undead) within the spell's area of effect. The bigger the area, the better off you will be. By the same token, something with a Night-Eye effect should be standard fare. You probably won't be able to brew potions until you've improved your skills a bit and you probably won't be finding items that have the effect until you're somewhere between level 5 and 10, but not having to carry a torch in order to see where you're going will make your lift sooooo much easier.
Carry at least a few Repair Hammers at all times. This is especially important if you are still a Novice in Light or Heavy Armor. It won't do you a lick of good to have that Mithril armor if its protection is reduced to zero in your first fight. I once stumbled onto Umbra as a 1st level character. Naturally I ran like hell and kept casting healing spells all of the way to the exit, but I didn't have a single point of armor left on anything by the time I got there. Sometimes I think this is one area where pure spell-slingers have an advantage over the rest of us - they don't have anything that needs repairing.
No matter what your class is, you're going to encounter locked containers. Spell-casters who are strong in the School of Alteration can work around this with Open spells. Sneaky-types will probably have a decently high Security skill and access to Thieves Guild fences where they can buy lockpicks. Those with the Tower birthsign get to open any pickable lock, but only once per day. For the rest of you, you're going to have to improvise. Those 50 or so lockpicks that you got from the Tutorial dungeon won't last forever. Consider searching out Nocturnal's shrine so that you can get the Skeleton Key (basically a reusable lockpick that boosts your Security skill by 40 points). As an alternative, Goblins and Bandits usually have a lockpick or two on them, but at one or two each, it will take a while to amass a stack that will hold you for a while.