The first thing you need is a shelter to do some night time chores safely. If you managed to find a herd of sheep and got at least 3 wool from them, build a bed before night comes. when you have bed, you don't need a shelter, you can place the bed anywhere on the ground and sleep on it as it is. But you may want to fall asleep as soon as the sun sets.
Simply watch the sun and when it touches the horizon, right-click the bed. This way you prevent any mobs from spawning and you are 100% safe from them. Though you may find some mobs around 60-120 blocks away from you, they can't attack you during the night.
Initially, your sole light source in caves and at night will be torches. These are made from coal and sticks. You probably don't have any coal yet, so you will have to resort to using charcoal. If you do have coal and torches, you can skip the next steps.
To make charcoal, you have to smelt blocks (logs) of wood. To do this you need a furnace. To create a furnace, arrange eight cobblestone blocks in a ring on your crafting table.
After you have placed your furnace, right-click on it and add a fuel in the bottom slot and the wood in the top slot. At this point your fuel of choice is probably wooden planks, two of which will smelt 3 items (logs, meat, ore, etc). Once you have your first pieces of charcoal, switch to using that as fuel -- each piece will smelt 8 items (just like mined coal), and you have other uses for planks.
Acquire about 6 pieces of charcoal. Each piece of charcoal (and coal, they are equal) can smelt 8 blocks, so it pays off more to use them, rather than wooden planks as a fuel. Now you can cook your food. Ration out a bit of charcoal for that and turn rest into torches.
Now, the final thing you need is a bed. You make a bed by combining 3 blocks of wool and 3 blocks of wood in your crafting menu.
While venturing out in the middle of night is an option, try to avoid it as much as you can. If you do not have enough wool to make a bed you may have to do so, but take extreme care. Wool can come from two sources: Sheep drop wool when killed; once you have a little iron for shears you'll be able to get the wool without killing them.
If sheep are not to be found, it is also possible to make wool from string, which you can get from killing giant spiders. You will need 12 pieces of string to make 3 pieces of wool. You can also get string from cobwebs, but those are only found in underground structures which are rather dangerous in their own right.