Drywall – What are the benefits of purple drywall

This will likely be a location specific issue, so it doesn't hurt to check your local codes.

All the fire rated work that I've done has been between units (both to other units and common areas) and any load bearing structures of a multi-unit dwelling. When we do that, fire blocking is installed in the ceiling space as well to prevent a fire from traveling through the ceiling to another unit. For everything else within a single unit, and in a single family home, we don't use any fire rated drywall.

Edit: A few reasons I'd disagree with Eric on this one. First, I'm presuming you already have a certificate of occupancy (or someone did) to move in with an unfinished utility room. That required an inspection that presumably allowed the space as is. Next, most codes for combustion devices (furnace, gas hot water) require that you have ventilation. Installing a vented door is perfectly acceptable and won't provide any fire stopping. And, finally, most HVAC's (if the furnace is forced air) will be connected to vents that run to every room in the house. From the little I saw, the furnaces and utility rooms in searches likely referred to multi-unit dwellings, and that does need to be fire stopped. You also have to use fire caulk around every hole. Our multi-unit buildings also have sprinklers throughout and spring loaded vents that snap closed in event of a fire.

That all being said, there's absolutely nothing bad about installing fire rated material around the furnace, other than the extra cost and that it's a bit more difficult to cut. If you want to do it, then by all means, do so.

Edit 2: One last note, realize that fire rated drywall increases the resistance from 30 minutes that you'd typically see with 1/2" drywall to 1 hour. Fire rated doesn't mean fire proof.

First, handle drainage. If you want to install a sump pump, perimeter drain, or water proof the walls, now is the time to do that. You should also install any plumbing drain lines at this point.

Next, framing. Concrete transmits moisture, so use pressure treated, and a styrofoam underlay that would normally go under the sill plate, to keep the walls dry. With pressure treated wood, you need galvanized nails to avoid a chemical reaction that would eat a normal nail. Make sure the framing is designed for the drywall (16" OC studs, nailing edge in the corners and around the ceiling perimeter). And make sure the walls are positioned/sized for any plumbing or other utility lines. For exterior walls, the thicker the wall (2x6 instead of 2x4) the more insulation you can include.

At this point, utilities go in, including plumbing, electrical, hvac, and communications. I'm a strong believer in conduit for running communications lines (like ethernet, catv, and phone) so that you can upgrade those without opening the walls or running wires all over the floor. Note that with more outlets, you will need more circuits, and copper isn't cheap.

Insulation is done after all the utilities are finished. If you have water proofed the walls, then I don't have much against the standard fiberglass insulation. Actually, I like fiberglass because it's easy to work with as a DIYer, relatively cheap, and replacing a piece or temporarily moving it (e.g. in the attic) is simple. However, with moisture concerns, or if you live in a northern climate with harsh winters, then I'd favor the spray foam. The spray foam has the downside of being difficult (impossible?) to remove, but it functions as a vapor barrier and leaves no cracks that air can get around.

After this, you have drywall, paint, doors and trim, and touch-up paint. I like to paint the walls and trim separately so there's less work on edging.

As for the flooring, you would usually do this before or after the trim. With hardwood, you would often do the flooring first, and then the trim goes up tight against it. With carpeting, you would install your trim 1/2" above the floor and the carpet installers would install the carpet right under that.