1. To carry something heavy.
2. To carry something in a dragging fashion, as if tired.
3. To go somewhere, particularly somewhere far away or otherwise difficult to reach; often implies resentment of putting forth such effort.
The elevator was broken, so I had to shlep the TV set up five flights of stairs.
I shlepped my book bag behind me.
I shlepped all the way out here from downtown so you could tell me you feel like staying in tonight?