OK - I would say you want a decent non-foaming cleanser, a salicylic toner, a basic moisturiser and SPF for day time.
Stop washing your face with shower gel - it doesn't like it. The shower gel will be too harsh for your face and stripping all the natural oils off and upsetting your face. This is also why cleansers shouldn't foam.
I've got oily skin and so can tell you nothing about hot cloth cleansing/oil cleansing/making your own cleanser/balm cleansing etc which lots of people are fans of. I am also too lazy to do any thing that takes too long. You don't need to double cleanse, neither does your face need to be so clean it squeaks.
Cheap option - Cetaphil. Slightly nicer options but still in the bland category - Lush Ultrabland, Avene Gentle Cleanser.
Next stop - they almost certainly are not blackheads, at least the ones on your nose aren't. They are sebaceous filaments and as you have noticed you are never ever ever going to get rid of them. Your pores naturally make a filament - you can squueze it out but by tomorrow there's a new one. The tops oxidize and turn black so they look like blackheads.
Do not lose hope - a salicylic toner will dissolve in the oil and exfoliate off the black bits. You still have the filaments but you can't see them. Given you already like Clinique, I'd go for Clinique Mild Clarifying Lotion - do not touch any of the other toners with a bargepole though!
Next moisturiser - this bit doesn't matter that much. It should be your smallest spend. Pick one you like
I am obsessed with Cerave.
SPF - you need UVA and UVB protection summer and winter - again going with the Clinique I'd go for City Block.
One you've done that and are happy with it, if you really want to do line prevention and glow then you are missing retinol and serum. For serum the serious science on the market is with Hylamide/NIOD. But real line prevention would be from a retinol like LRP Redermic R. There are a few more bits and bobs but 99% of line prevention is SPF, retinol and acid (in your case the salicylic)
Anything else about stem cells, magic serums, essences or oils is just beauty company gubbins basically. Also look at your pores from the distance other people look at them - suddenly they look like normal pores 