You'll be in one of three places: a little cave, a big dark outdoors place, or an "island" (basically your transporter) in the middle of water. Wherever you land, the first thing to do is get to safety!
If you're on the darkened plane, dig down several blocks and then dig out very fast. Be prepared to beat off mobs (luckily, most of them show up on the minimap.) Mobs will come after you, and if you're on level 80 or above, you get different types of explode-y creepers,
If you're on the island, extend the island using your leaves (you can only extend it one block....try for two blocks over the water and you end up in the water). On harder modes, use the leaves to build a "fortress wall" because everything out there is coming for you right away. Once you're safe, build a boat to go to the "mainland" (if you die while doing this, it's almost impossible to retrieve your stuff from the bottom of the pitch-black water environment. Reset and try again.)
If you're in the cave, bring out a cube of dirt, plant one of your saplings on it, and relax. You're in the clear for now. Start digging outward and keep looking on your map for a patch of dark blue, indicating open space and water. You'll want to head there to make your home.
Torches work differently in Descent to the Core than they do in Minecraft. While they are your friend, they also set things on fire. Keep them away from trees, vines, coal, netherrack, and small children passing by.
Vines are a good source of oxygen (particularly if you have a limited number of saplings) but you'll need to keep them 5 blocks away from torches.
You can starve to death. The two easiest foods to get are "rock soup" (probably hard on your teeth) and "Limes" from limestone. Rock soups do not spoil, but they do use up wood and stone. Limes are also your best and fastest growing crop.
Don't get too fond of her. She has a habit of walking into walls and dying if she's penned up, and running into attacking creepers and zombies if she isn't penned up. I stick her behind stone walls, but that's not a guarantee. The first quest under "Noah's Ark" gives you another sanity pet, Rhea... but you have to find and kill some spiders, and if Gertrud's dead by this point, it's going to be a challenge.
STUFF TO COLLECT
Keep any marble and limestone you find. Eventually you'll need two stacks of the stuff. You'll also need two blocks of compressed gravel eventually.
Mine all the aluminum and copper that you see. Copper can be used for armor and a lot of other things. You won't be able to mine ferrous ore, iron, or anything else for a very long time.
You will need to find lots of dirt - and that's found in the open areas where all the monsters are. If you're not in the open area, 9 rotten foods (most of your food goes rotten) will make one dirt. Fastest way to get spoilable food is to create limes from limestone (it takes 4 days for food to go bad.)
Believe it or not, they (like hay bales and other stuff like enchanting tables) are buried in the rock.
Cheat mode is enabled, but that takes the challenge out of it, I think. The one hack I do use after I get tired of dying is the map teleport (if you type M to get the map and click on one of the X's, you can either eliminate them or you can teleport there. I save useful locations)
Keep a tree and a block of dirt on you at all times.
One thing I wasn't aware of when I was first playing is that your pickaxe (and other tools) upgrade as you use them... so repair (with one piece of flint) rather than replace. Use 3 gravel to make one flint to repair pickaxe. No need for repair table - type E and do it in your 4 box crafting slot (flint on top, broken pickaxe on bottom)
No matter how stupid the "blind" rewards seem, put them in an inventory somewhere. They'll be something you need for a quest turn-in later on.
If you die and lose your pick (in the deep water, for example) you will have to make another flint pickaxe and level it. It's not much fun. Don't take your pickaxe on the water.
Boats are surprisingly fragile. Running into the shoreline, baling out of the boat, zombies, and perhaps even harsh words will cause it to vanish.
Zombies have pickaxes AND TNT. They will come into your house and will let other mobs in.
There's a display in the middle left of your screen that tells you what "floor" you are on. You start out around level 50, so you have to go up for copper and up even more for iron.
Watch your inventory display ("E") - you get experience for some crafting actions. After you've accumulated a certain number of points, you will have the option to unlock another inventory slot.
Build ledges and pathways as you go, along with "hides" to duck into for those times when you discover that you're being shot by a skeleton (or worse) and need to get out of their sight. Watch out where you place the torches, though.