I live in Germany. While I happened to be working during the World Cup match between Germany and Brazil last night and was unable to watch the game, I was still able to keep up with what was going on, thanks to the honking and cheering from the street outside. While that particular street noise was fun and helpful, other outside noise can be terribly annoying. But there are some things you can do about it.
Use your exterior walls
Nothing absorbs noise like a fat wall of books. Consider moving your bookshelves to line your outside walls so the books will help absorb some of the street noise. Fabric is another good insulator, so a solution that moves your closets, clothing racks, or linen storage to the outer walls can help as well.
Get thick curtains
Heavy curtains can also help dampen sound. The thicker and heavier the curtains, the more effective they'll be. You can even buy special noise-absorbing curtains designed to help cancel out annoying sounds. While I would love to drape my entire apartment in velvet someday, for now I'm using IKEA's Werna blackout curtains. They're very thick and heavy, don't cost a fortune, and as a bonus they also block all the light that would otherwise be seeping in. If the noise makes you batty, you might even want to consider hanging two sets of curtains.
I find that a fan or a white noise machine helps a lot.
Reinforce the windows
A lot of noise seeps in through windows. While you're probably not going to be able to convince your landlord to replace your standard rental apartment windows with better ones that keep out more noise, there are some DIY options you can consider. Covering the windows entirely would probably turn your apartment into a dark little box, but cutting some sheets of acoustic foam to fit the windows will give you an option you can put in place at night and then take down whenever you want your windows back.
I like to save the squishy little foam earplugs that airplanes give out on international flights. They're comfortable enough to fall asleep in and keep out almost any noise.
Trust time to fix it
In the end, time will probably take care of most of the problem for you. I grew up in a major city, so sirens and car horns don't even register as noise for me. When my husband first moved to a city, he couldn't sleep for weeks because of all the noise. He was convinced he would never sleep again. But after a short time he got used to it, and now he can sleep through anything. He even slept through the World Cup game last night, and that was so loud I thought the windows were going to fall out. Give it some time and you may find you don't need any noise-canceling solutions at all.