I am recently starting to DM a new game, and it's about 2 sessions in. Any ideas that are exciting but not too hard for the players! Like to think outside the box. 

It depends on what you have planned and what your group might be interested in as well as what background information you have on their characters and their wants and desires.

Any information on what you have planned already or maybe party composition or something for us to get an idea of what to suggest?

A fun one is a wizard tower filled with traps, puzzles, riddle, mimics or feats of strength (or any other ability really). 

A fun diplomatic encounter I've run is two party of goblins at a stalemate because there is a gold piece in the middle of the road. 

City under attack.  Whatever city they are in  gets attacked by a horde of monsters they haven't faced yet, Like undead controlled by a Necromancer, or a mad mage sends a group of low level elemental. This allows the party to really show off what they can do, and role play saving towns people.

A king or queen is possessed by a evil force, A hero stays behind with him to try and keep it from making bad decisions for the land it rules over, And watches him or her become more sick, While the rest of the heroes try to find the source of the evil, Thats controlling the king or queen and put a stop to it.

What I often do is to determine what I see as the "end scene" of the adventure, then work backwards. For example, I decide that the end scene should be a fight. Since you're starting out, a CR 2 monster makes a good villain at the end. I filter the monsters and decide to go with an ogre. Since Ogres tend to be a little dumb, I decide he's actually the lackey of a Mage that the PCs won't encounter yet.

Maybe the Mage wants a magical item, and it's using local muscle (goblins or orcs perhaps) to try and get it. The Mage has set one of its minions, an ogre, over the muscle, and it has them hunting ruins that are too close to a town for comfort...especially if the lazy goblins keep stealing food from the town.

now I have a hook (goblins are stealing from us...help!), a setting (the ruins), some minion types (the goblins), a boss fight (the ogre), and a possible recurring villain (the Mage).

Aethan uses a really good method for when you don't yet have a direction yet for your players. i myself have done similar things creating the end point first and then coming up with an interesting path that matches where they are going.  Also as he mentioned you can use events for giving your players a lead on what else might be afoot in the realm.  Hints to a power behind a small encounter or that it is symptomatic of something coming (plague, an army, civil unrest) in the future.

The other side is do you have any plans for situations you want to put particular players in?  Has your cleric been feeling a little ineffective lately?  Perhaps give him a chance to fight some undead enemies and be in the spotlight a bit (or even be the driving point for the events).  The barbarian has just been getting murdered by the big enemies, so you let them deal with a horde of weak creatures that let them smash them with wild abandon.  The opposite side of this coin is the players have been cruising through scenarios so you throw them a curveball, like a charisma/diplomacy event in a group that is very weak with charisma.

So I guess my thoughts on the matter are as follows.  Start from the end point and build around it.  Create an event to draw them into larger goings on in the area and build around that theme.  Focus on a character and create a story either revolving around them, or giving them a chance to shine.

One thing I really enjoy is to have the campaign happen some time after a previous one I'd run or play in, and any odd ends left unfinished may have developed into their own issues that can draw in a new group of heroes, completely unrelated to the prior adventure but still connected. Even more fun with a constant group of players who are into that style. 

Personally, I’m a fan of the Hex-Crawl style of campaign. It encourages an emerging story style of gaming that gives the player plenty of choices (about 6 of them in fact :) ) while giving the DM the flexibility to cultivate some plot lines based on where the players go. 

Be sure mix up encounter types. All combat makes boring play. And to encourage the roleplay, be sure to reward that just as well as combat, if not better.

For four or more new characters, a boss at CR 3 or 4 is not too much, if they proceed cautiously and wisely. How about a new local 'lord', who takes over and refurbishes the old keep a mile out of town. Now he's forcing the village to pay taxes. Some roleplay, a few guards, a trap, and a boss fight should fillan hour or two. Perhaps even scale up the number of guards or encounters to push the characters to the next level.

Good luck.

The characters are wanderers hired to solve a minor issue in a shady settlement and end up rewarded a residence on it and worship among its people once the quest is done, only to find out that the town/village/city hides more than a few mysteries and conspiracies of both the common and supernatural kind.

This starts the campaign with a simple dungeon crawl, sets up several possible world-spanning hooks and gives them the opportunity to role-play their daily lives and misadventures between the adventures, which I honestly think is something more campaigns should do!

Thanks for these suggestions. I'll be talking to my table about them as well.

DM: You start in a tavern with your party of close friends...Without warning EXPLOSIONS and FIRE obliterate the tavern. What do you do?

start a background story first(how did everyone meet) and then build from there.

I like a simple "rumor of treasure" to give the players an opportunity to fight some low level monsters, fund their adventuring further and learn the lay of the land in the area they begin.  The treasure is appropriate for level 2 or 3 characters, with a magic item for each one and plenty of story hooks thrown all about to see what they want to do.  The best treasure is they meet some NPCs they can ask for help later.