I played a few hours the other day, and I basically didn't know how to build a house before night, and ran into trouble a bit. But it seems to get easier after respawning the world a few times and reading a few things. And getting used to the controls, interface, and lack of a pause button... Crafting with a bad interface and no pause button while being a n00b hurts a lot. :(

Does trouble come looking for you later on, or do you have to explore to encounter new perils? I know that Don't Starve, for example, will throw Winter at you, and monster attacks start to occur after a week or two, and even later on epic boss monsters spawn. What about this game? And can you just keep spawning new worlds and taking your loot with you to avoid it?

1. "Does trouble come looking for you later on or do you have to explore to encounter new perils?"
A bit of both but there are non time related triggers for when it comes looking for you. Basically unlike Don't Starve Terraria doesn't throw anything at you that you shouldn't not be ready for in terms of health and gear, unless of course you trigger it yourself through a summoning item or wandering in a area and doing something that will summon a boss such as while exploring the corruption/crimson biome if you break three corruption orbs or crim hearts (orbs and hearts drop items when broken) but it will spawn a boss, the game does hint that something will happen if you keep doing that by throwing messages like "you feel a chill run down your spin" and "screams echo around you" but again it usually warns you that you may want to make sure your not getting over your head. Outside of that events like the Blood Moon and Goblin invasion will only start occurring once you hit a certain amount of health and have set amount of defense.

2. "can you just keep spawning new worlds and taking your loot with you to avoid it?"
Not really, while you can jump from world to world since the triggers are stat and item related it's only a matter of time before you trigger them in another world. So you would be jumping so many times it would be pretty stupid. Also you would be missing out on being able to get better gear, for example a certain area of the game will only be explorable if you kill it's boss, if you try to enter it without doing so you will be killed, also to get a certain type of ore(in decent amounts anyway) you have to kill two bosses, Blood moons only last one night and the worst that happens is it spawns hordes of zombies and eyes and zombies can break down doors to your house but if your underground and can't actually see your house you have nothing to worry about. Goblin invasion is a diffrent story since they will always spawn where you are but the game will only spawn goblins and they aren't that bad.

You'll have to explore. During night stuff can happen but a beginner tips is to trap the Guide NPC in some ground where only you can access by digging down (so he can't escape left or right). After that try to build a house.

And avoid spawning new worlds as you will explore and unlock secrets that will be dependent on that world and you'll destroy the charm of exploration otherwise. If dying is an issue, just play softcore and all you lose is money which you can drop off in a Chest close to the Guide or in a house after you've built one everytime you return from an exploration somewhere.

Yes trouble does come looking for you.
It will either be via shots that phase through solid walls or an enemy that phases through walls or even teleports to you.
At the moment i have a start up character at a depth of 2000feet and have had two rare spawns of an undead mage crop up (teleports to location & fires 3 phased projectiles).

Basically unlike Minecraft you can't sit back and autofarm while AFK in Terraria. At least not without building a 12 block gap filled with lava around a fortress that you've funneled about 12 characters into 1 space as meat shields for you.

As has been posted progression events are tied to health etc.
In particular though don't rush to hard mode by breaching hell till your confident in the game; plenty of stuff to keep you busy and entertained till then.

Cool, the game sounds sort of fair about this kind of thing. But it's good to know that there are also triggers, and that it's also reaaaally not worth it to take on tough challenges early.

Just found the magic mirror; now I can explore all I want, day and night.
God I love this game. :D

yeah magic mirror comes in handy, just whatever you do NEVER USE IT WHEN FIGHTING THE WALL OF FLESH, even if your going to die in the fight just take the defeat because if you use the magic mirror to escape that fight bad crap will happen(last time I tried I exploded when I warped back home and then when I re-spawned it dragged me all the way back to hell). Bad things also happen if you try to escape Hell without magic mirror when fighting it, when you try to run out of hell once it(the wall) spawns it will drag you back down into hell and you will take a ton of damage.

The first boss will have a chance of spawning when you have 200hp or more so once you get to 180 don't use another heart till you feel ready.

Surprised no one else has mentioned it but there is a pause of sorts ... in the options you can turn on auto-pause and what that does is every time you open your inventory the game is paused. Personally i play with this off because then you can build with your inventory open i/e take items/blocks directly from your inventory and place them in the world while holding them on your mouse cursor. :)

Pretty much all of the jumps in difficulty are player triggered, though you may not know that you've triggered them. As mentioned, the first boss will start to randomly appear once you have 200 HP (the wiki adds that you need at least 10 defense and 3 NPCs as well), invasions will only start once you have that HP as well as destroying a certain Crimson/Corruption object, meteorites won't randomly fall until at least one of those Crimson/Corruption objects has been destroyed, defeating certain bosses triggers other changes, etc.

Most of the bosses are player-triggered only, usually either with a useable item or by destroying an object, though some will randomly spawn (which itself needs to be triggered; and again, you may not realize that you've done so) until they are defeated at least once. AFAIK, all of the randomly spawning ones will only show up (and can only be summoned) at night, so even when you've triggered their random spawns you're still safe during the day. You'll also get a warning before they spawn, at least, so you can warp home and dump your stuff in a chest if you're worried about losing anything. There's one exception to all that, but... well, I don't know if I'd really call it a boss, despite the music and announcement, and I've only ever seen it 3-4 times in hundreds of hours of play.

I found it quite helpful to build myself a small tower a short distance away from my base and surround it with lava. Sit in the tower and watch as the invasion kills itself in the lava moat (well, except the mages ;))

Actually there is a trick to invasions too.
Invasions happen within a certain distance to the map's original spawn point (at least the goblin invasion does).
So while you can't prevent them from triggering, you can make sure your town (or whatever you call the accomodations for the NPCs) is safe, and you can terraform the zone around the original spawn to make sure everiything can be easilly reached and/or killed.

The game's pace is the pace you decide are comfortable with. This is a gift of the freedom in Terraria in regards to being able to build your own buildings, bases, and shaping the land to suit your needs.

It helps to build a small house, even if it isn't very fancy on the 1st night which can serve as a temporary shelter to protect yourself from zombies and eyes. The 1st boss can potentially spawn when you get 200 health or more. And when hardmode begins, there is a small chance of a hardmode boss spawning every night. If you're playing on softcore mode which I'm assuming you are, at the point you're strong enough to face bosses, dying is just an inconvenience where you can lose money.

Aside from those things, you decide when you want to use the ore and items you collect and how to use them. A great philosophy when playing Terraria is "You don't adapt yourself to your world, you make your world adapt to you" It sort of fits with how you can dig and build however you want.