@jjj70095 , The questions are pretty open ended. You can get a whole variety of answers. It may help to narrow down some details.
First, I'd like to know why a new furnace is causing moisture? Are you getting condensation inside the glass or is it forming on the surface? If it's multi pane glass with condensation inside, that's a failed glass unit. If the rest of the door is fine, I'd just get a new sealed glass unit.
Condensation on the interior surface of the door is relative to two details. Moisture level and temperature variance. When the surface reaches dew point, condensation forms. It may be a poor performing door assembly, high moisture level, or a combination of both. What is your indoor humidity? If your new furnace is considerably more efficient, you are running less high heat cycling. That will dry the air less. We are also on the tail end of a warm spell that hasn't forced long heat runs. That means less moisture is forced out of the air. Humidity levels inside homes are higher than typical for this time of year. I'd love to know your current indoor humidity.
I know that is a lot of questions that don't even address the simple details of price and how long it takes. On the other hand, I've seen plenty of doors and windows replaced without ever addressing the actual problem of too much moisture. Most of those projects are money buckets.
Replacing a door varies a lot on building cladding and the integrity of the opening frame. You can screw and caulk a great door into a junk opening for little money. Entry doors can be spendy ventures. It looks like you want to go with a basic door and save the big bucks. I'd first verify that you have a standard opening size. It's usually simple enough to measure the slab just to get an idea. 36 X 79 is a common slab size for entry doors. If you have a spec size door, purchase and installation are generally less expensive. Simple exterior casing (trim) also helps with those expenses.
We could go on and on with approximate costs, but It'd be shooting in the air without some more details.