1) Pit Fighting - as others have said, make your dwarves fight the goblins. Put the animal stockpile 3-4 levels above your barracks. Channel a drop from the animal stockpile to the barracks ceiling and assign it as a pit. Drag gobbo prisoners over and drop them down to be stunned for a moment before getting dwarf-piled by your military. Give your military wooden training weapons and imagine the carnage of the equivalent of an unarmed goblin getting beat to death with baseball bats.
2) Trade goods! You don't have to piss off the humans or elves if you don't want to. But you piss off the goblins by not being goblins, so they will attempt to slaughter you whenever possible. Assuming you keep the peace with humans and elves (*spit*) , you can use all those troll fur robes for trade goods and buy out entire caravans of more useful stuff like metals, food, wood and cloth. Even if you decide to tell the elves and humans to "stuff it" and let them attack, you can ask your liaison to bring all sorts of steel items that you can melt down for weapons and armor yourself that you will be able to afford with several crates worth of cave spider silk socks.
As others have said, though, there's a certain "Dwarfiness" associated with Dwarf Fortress that involves catching goblins in cage traps and pitting them against your fighters for training, pissing off elves by slaughtering the lot of them and cutting down every tree in sight, and chuffing off the humans just for the sake of seeing if you can withstand a human siege.
In Dwarf Fortress, you really can play any way you'd like, but for one reason or another, the community has developed along with the feel of the game such that we very much approve when your dwarves sieze caravans full of useless wooden stuff, drop the elven traders into magma, then piss off the next elven caravan by trying to trade them all the crappy wooden junk you just stole from the last caravan. And raze the surrounding forests for good measure.
And human sieges are a true trial of dwarven fortitude, since they actually siege and don't just rush your fortress entrance to blunder into all your carefully-planned traps. If the human diplomat has been in your fort, he knows where all the traps are, so tough luck trying to get the army to stand on them. Plus, they will sit outside your fort for seasons, not doing anything, until you finally do something to break the siege.
There's no reason to play like that if you don't want though. There's nothing wrong with building an entirely above-ground fort out of stone so you don't have to clearcut entire forests for the wood to build the little cottages and trade peacefully with everyone you can. (Goblins will still attack you though.) Nothing at all. Nope, nothing horribly, awfully elven (*spit*) about that at all.
And we won't point and make fun of you.
Most of us anyway.
Much.