I am Reform and have never davened privately but am considering davening more often. One thing I am confused about is what to do with tunes (nusach). Are they only for public worship, or is it necessary/preferable to chant the prayers even when praying alone?
The OU has good advice on this topic
Say prayers (and blessings) out loud, but not overly loud, in order to help you to concentrate on what you are saying.
Exception: The main exception is the amida prayer, which is said quietly enough that you can hear yourself but that people near you cannot hear what you are saying.
You should feel free to use the tunes (nusachim) if they help you concentrate or raise your enjoyment but they are by no means obligatory. Even in public prayers, there are certain minyanim that will only use minimal amount of chanting.
See also this related question on whispering.