I too suffer from dark circles, but skincare generally won't help with that. The Maybelline age rewind (I think it's called) concealer with the little sponge on top is great for covering dark circles and has a few skincare benefits as well.
If you have the odd spot and dry patches I'd definitely recommend acid toning. Salicylic acid is good for pore-related issues, like spots and blackheads, and glycolic is for resurfacing, so would take care of your dry patches. Nip+Fab do Dragon's Blood pads, which contain salicylic, and also some glycolic ones, and they're usually in some kind of offer. I alternate between the two.
For cleansing, do you wear much make up or SPF? If not, you could probably get away with one cleanser, like the Superdrug Vitamin E or Natural Radiance hot cloth cleansers. If you wear make up I'd recommend double cleansing, using the above as the second cleanse and an oil as your first cleanse. Some people just use almond oil or similar, but the Body Shop, Una Brennan in Boots and L'Oreal do good budget options.
A budget moisturiser I'd recommend for pretty much all skin types is the Dirty Works (available in Sainsburys) 8 in 1 Miracle Cream.
I personally use a lot more products than that, but I am mental a skincare enthusiast. I think it's good to try a few things out first in case of reactions, but you may want to think about a serum and/or a facial oil later.