You will need lodgings, but the lab rules will be based off the Stronghold Builder's Guidebook. This is not the final price, but will give you an idea what I'll be looking at.
1 ss is one stronghold space. About 20x20x10
Alchemical Laboratory, Basic
Size: 1 ss
Cost: 700 gp
All sorts of vials, flasks, beakers, burners, crucibles, scales, measuring devices, and other alchemical equipment fill this space. The room also includes a pair of basins and a ready supply of water in a barrel or two lining the wall. Along one wall stands a fireplace that not only heats the room but also provides the fire necessary for so many kinds of alchemical recipes.
The floor is rough wood or stone, often stained with chemicals. Shelves stocked with chemicals and alchemical reagents line the walls.
Using this lab gives a single character a +2 circumstance bonus on his Alchemy checks.