You HAVE to oil cleanse, or you have to give your AB badge back! 
Honestly, you won't look back, plus it's the only way to properly remove sunscreen. At the mo I'm just using this one and it does the job, emulsifies beautifully as well.
After taking makeup off with oil you need to properly cleanse but try to make sure it's a low ph cleanser. This will enable your actives to get working straight away plus it's just healthier for skin.
I use 3 cleansers and I alternate between a Konjac sponge, and a bubble net, which I love. My cleansers are Cosrx Low Ph good morning cleanser and a Cerave low ph cleanser. I also have a low ph Sulwhasoo cleanser which is wonderful, but expensive.
In the evening I double cleanse, in the morning I just use the foaming cleanser. I also swipe my face with a low ph toner(Mizon) post cleansing and prior to starting my routine.
This is a great post about cleansing.
I'm using the Cosrx actives you mentioned and they work well for me, I also sometimes use Stridex for my BHA and some Nip and Fab pads for my AHA, just depends what mood I'm in.
The Cosrx BHA cleared my daughter's acne, it's very gentle as it uses a different BHA than Salicylic acid which is super kind to the skin.