Are you near an Apple Store? If so just make an appointment and they can do it pretty quick in a lot of instances. As far as sending it in via mail I am not sure. I will have to let another comment on that as I am not too sure. Also there are third party places that can do it but do your research first.
If you're sending it in, it can take approximately a week depending on business days and or when they receive it. But if you are having the repair completed in store, which I would suggest, it's a much quicker method and you can have it within the same day.
And yes, Apple does a good job with their repairs. I only use them over any other repair service.
Always do a encrypted backup including your images BEFORE you go to the Apple store or mail it in. They might replace your entire phone. Quality of repairs is up to the person doing the work. Note if their are any scratches on your device before turning it over.