This can be a very complicated subject. I've done a lot of work with non photosynthetic/predatory corals and invertebrates and kept a lot of very demanding species. To start, every coral polyp has a mouth, and is able to ingest food, that's what they're there for. Many of the "normal" corals we keep aren't too picky There is a lot of research into herbivory in corals, as well as other types of feeding. Herbivory as in feeding on phytoplankton, any reef system will benefit from it. If you aren't keeping filter feeders or anything really demanding then a good mix of phytoplankton species and small-medium (5-300micron) particulate foods will work for most reef systems. I like the golden pearl diets and used them plenty in the past. There are also many coral foods sold that will work.
Feeding schedule will differ. If you're feeding phytoplankton you can do it nightly, powdered foods daily-3 times per week. If you're spot feeding with larger foods to bigger polyps (mysis shrimp, krill pacifica, things like that) it can be two times per week or weekly. I've kept non photosynthetic reef systems that were fed by an automatic doser every hour and the water was always slightly tinged with food. It's up to you and which corals you'll be keeping.
For what to feed with, seriously buy a Turkey baster, every reef keeper needs one. Also little pipettes and syringes work too I use them often.