Since you haven't been using it that long you presumably don't have a ton of personal data stored on it yet. I'd take the opportunity to back up what you need, then do a full factory reset on it and start all over again.
It's possible one of your apps is responsible for the slowness so try reinstalling them one at a time to see if you can figure out which one it might be. Also keep an eye on Settings/Battery to see what is using the most battery, sometimes a misbehaving app or service will show up near the top of that list. Apps like GSam Battery Monitor and System Panel can help you see what's going on as well.
Be wary of apps that are constantly running in the background and using data, sending push notifications etc. as these can slow things down. Look at settings, turn off things you don't need. If you have any "task killer" or "battery saver" apps running get rid of them, they actually make things worse.
What makes you think the phone wasn't actually new? Was the store a reputable one or not so much (i.e. mall kiosk or guy in white van)?
