It could mean to continue something. Example The TV series is over for this year, but the station plans to pick it up again next season.

I would have to see the context to tell for sure. To pick up again I would take to mean start over again or to move again. When you 'pick up stakes again' it would mean to pack all your belongings and move. It comes from picking up tent stakes.

I have never heard that phrase. You may be thinking of "pick it up" which can mean the obvious -- someone wishes you to pick up an object. It can also mean "to increase your efforts or energy in a given task." For example, if you were running and want someone to run faster, you might say "let's pick it up." The "it" in this case would refer to the pace of running.