You're handing the music off to a composer, are you handing the sound editing, sound design and mixing off as well? If so, there is not much more you need to do besides getting all the production sound and other recordings organised and making sure you know how to create an AAF or OMF, ready for delivery to the audio post person/people.
If you're not handing it off then what you are asking me to do is explain how to do three significant filmmaking roles, not something I can do without writing a book or two! My strong advice is to give the job to someone who knows what they are doing with sound, I'm not sure there is any excuse for not doing this!!! If you feel you absolutely have to do it yourself then the first stage of audio post is the sound editing, so you should start there. First thing I would do is to get rid of all the editing clicks by re-editing on zero crossing points, get rid of all the abrupt cut offs by extending the ends of the audio clips which are cut off. Go through the dialogue, a word at a time, editing out clicks, pops, mouth smacks, etc. Also part of the editing process is to sync appropriate SFX and Foley. What is "appropriate" is your decision but there has to be some logic to it. For example, you have a scene where someone is getting punched and is then thrown on the ground. We hear them landing on the ground but not the punches, grunts or any other sounds, there's no logic to that. Just having a few SFX or bits of foley stuck in with silence in between won't work. For those scenes where you are going to have sound, rather than just music, you will have to record/edit in/construct ambiances, background SFX and Foley to create the illusion of a real acoustic space. You will also have to use your judgement of what is "appropriate" with regard to the abstract SFX you design. For example, you use a reoccurring sound effect near the beginning which sounds like an unprocessed cheap cabinet door being opened. I personally wouldn't call that appropriate. Once you have the editing sorted out you can concentrate more on the sound design, before you finally move on to mixing the whole thing together.
G