Pertty simple realy…How were carrier/messenger pigeons trained?
I understand homing pigeons find their way home navigating by the sun but how do they know where to deliver a message to before returning home?
They only know how to get “home”.
So if you were (for example) in some Napoleonic War outpost and needed to get a message to HQ, you needed a pigeon who knew HQ as “home”, and you needed an HQ that stayed in the same place. But the same pigeon could not be used to send a reply - that would require a pigeon that knew your (fixed) location as “home”. Needless to say, distribution of pigeons around the battleground could be both bothersome and dangerous, so the advent of battlefield radio was looked on as “the greatest thing since sliced squab” 
The place they deliver the message to is their home. The birds are conditioned to think of a location as “home”, where they nest. They are then taken to the location where the message is to be delivered from. When released, the pigeon returns home, carrying the message. In the case of “two way” messenger pigeons, one location is where they are conditioned to nest and find water, the other is where they’ve been to taught to find food. Precisely how they accomplish their feats of navigations is the subject of some debate, but it is believed the birds have some kind of magnetic sense, which they use in conjuction with sun position and landmarks to find their way.
My dad has had some homing pigeons for years, they are fairly easy to deal with. He has them for a hobby and for dog training. For dog training we can put them in a (remotely controlled) bird launcher (to help train a pointing dog), and when we release the bird out of the launcher (it throws them up about 5 feet in the air, then they fly), they conveniently fly back home and we can use them again next time. No hard to the bird, and we can use them over and over because they just fly back home.
We start after they can fly well, and maybe take them a half mile away. We then double the distance the next time we fly them. We have some that can go 80 miles no problem - haven’t really tested them beyond that because we usually don’t drive further than that from home on a routine basis.
How long does it take, typically, before a new bird is trained to its new home location?
A recent TV show featured 4 types of birds, one of which was the homing pigeon. A company out west that runs rafts down rapids uses homing pigeons. A guide that is standing on a rock takes pictures of the rafts as they pass. The film is then placed in a harness on the bird. The pigeon then returns home, the film is developed and the raft passengers have pictures of themselves available upon arrival at the end of the trip.
For the more common one-way pigeons, I don’t believe they are typically “trained to a new home location”. My grandfather used to race pigeons, and he would either breed them himself or buy them as babies. They grow up pretty fast so they can be fledged and trained within a few months. If you try to transplant an adult, they might bolt for their former home the first time they’re set free. There are some web sites that suggest that it can be done, but I don’t think it’s all that effective relative to raising babies.
Regarding “two way” pigeons, I hadn’t been aware that there was such a thing until you brought it up. I think they’re not very common. This site suggests that they can be conditioned in about a week, but that they’re higher maintenance after they’re trained.