I am flying to Bali in March and am trying to book directly with United. I would like to fly through Singapore on the outbound from Houston and then fly from Bali to Sydney and spend a few days in Australia on the way home. (United will have directs from Sydney to Houston by the time I travel..woohoo)
My question is if I purchase IAH to SIN and SYD to IAH, then separately purchase tickets in and out of BALI on smaller non alliance carriers will I have any trouble in customs, or with my luggage?

Any advice you can give would be appreciated!