Since this topic comes up quite a lot on this board I now keep a standby document which I've copied below. Usually requests like these are accompanied by a (budget) product suggestion request so I've left those in. As mentioned by pp start with double cleansing only (assuming you're at least using a moisturiser or something now) then start introducing new products slowly, and I stress slowly. We're talking weeks not days between products.
In regards to your proposed routine: You do not need to double cleanse in the morning, if you're normal to dry skinned you're often better off only rinsing with water in the morning and do this at the sink. Shower water tends to be too hot and too plentiful (though plenty do this without issue, I'm simply of the eliminate all irritating possibilities scenario when starting to minimise potential problems).
With regards to retinol, technically it's an acid and depending on instructions used in the serum stage or on its own. Be mindful that it irritates a lot of people and should be the last thing you should be venturing into when starting a skincare routine from scratch, everything else should be working for you before even pondering about retinols.
Masks and peels are extras, I only really recommend a simple clay mask to clean out pores. Beyond that it's mostly pampering and fun that bring temporary effects but contribute little to an established skincare routine.
So without further ado, the copy paste overview:
So before we get to the basics of a skincare routine, drink plenty of water (especially if you drink a lot of caffeinated drinks, they're only making you more dehydrated). Avoid sugar, alcohol, cigarettes, and everything else you know is shit for you.
The basic routine is usually something along these lines, double cleansing (in the evening, single cleanse/wash in the morning) - acid toning (optional) - serum/oil (optional) - moisturiser - SPF (daytime only). Full disclosure I'm not UK based (so no access to a lot of the UK pharmacy brands) and have sensitive skin so I'm no good with recommending budget products from personal use, I'm merely listing the products that are repeatedly recommended here on the S&B boards.
First Cleanse: You want an oil based cleanser (not to be confused with the 'natural' oil cleansing method) to cut through make-up and SPF which you remove with a warm flannel (despite the method being called 'hot' cloth cleansing, hot water is shit for your skin) or a muslin. Use a clean flannel/muslin each day. For mornings (or days when you don't wear make-up and/or SPF) skip this step.
Suggested products: Body Shop Camomile Sumptuous Cleansing Butter, Superdrug Vitamin E Hot Cloth Cleanser, Virgin Coconut Oil or Superfacialist Una Brennan Vitamin C Skin Renew Cleansing Oil. Mid budget - Clinique Take the Day Off oil or balm. High budget options- Emma Hardie Moringa Cleansing Balm, Eve Lom Cleanser.
Second Cleanse: The first cleanse was to remove all make-up and SPF, the second is to clean the actual skin and any residue. Gentle is what you want here, a milk or cream cleanser. Ignore instructions of removing with cotton pads etc. Simply apply to dry skin and take off with warm flannel/muslin from above.
Suggested products: Cetaphil Gentle Skin Cleanser, not an S&B but one I've used and fall back on often, Weleda Almond Soothing Natural Cleansing Lotion
Acid toning: The quickest way to kick start your skin again, caution is warranted. Slow and steady is the way to go here, introducing it once or twice a week and building it up to a point that feels comfortable for your skin which can be anything from twice a day to once a week/fortnight. AHA helps break down dead skin cells and encourages cell turnover, revealing more 'youthful' and radiant skin. BHA works deeper down and is good for cleaning out pores and with spots. There are more types of acids and both AHA and BHA are far more complex than described but this is their basic setting. I would only suggest incorporating acids into your routine when you have your basic setup of cleanser - (serum) - moisturiser - SPF in place. If you can't commit to daily SPF and regular top ups during the day in the height of summer do not bother with acid toning, it makes the skin photosensitive, you will only damage and worsen it if not diligent with SPF post acid toning.
Suggested products: AHA - Nip + Fab Glycolic/Glycolic Extreme pads, Pixi Glow Tonic, Alpha H Liquid Gold. BHA - Nip+Fab Dragon's Blood Fix Pads. BHA/AHA combo- Olay Anti-Wrinkle Toner.
Serum: A concentrated and light weight formulation that penetrates skin more effectively than moisturiser. This is where you address your specific skincare needs, hydration, acne, sensitivity, anti-ageing, etc. Alternatively an oil can be used in this stage following it with or without moisturiser depending on need.
Suggested products: Hydration- Superdrug Simply Pure Hydrating Serum, Vichy Aqualia Thermal, Acne- La Roche Posay Effaclar Duo, Anti-ageing: La Roche Posay Redermic. Oils- Almond Oil, Virgin Coconut Oil, Body Shop Oils of Life.
Moisturiser: Seals everything in and hydrates, for whatever reason we're still all stuck on the moisturiser being the most important and life changing product in your skincare routine. It's not. Probably leftover marketing from the anti-ageing overpriced cream era, moisturisers aren't going to do much on their own. Spend your money on serums and get a no frills workhorse.
Suggested products: Go with the one you like already, oft mentioned on the S&B boards are, Eucerin Hyaluron Filler, Body Shop Vitamin E, Aldi Caviar and Aldi CoQ10. Please note that day creams may have an SPF in them, they are by no means sufficient and to avoid clogging your skin unnecessarily do not use an SPF containing day cream at night, you will need to double cleanse it off in the morning. Needless washing results in aggravated skin. The easiest is to have one moisturiser without an SPF in it.
A separate SPF: Beyond its necessity for use with acid toning, it is the most effective anti-ageing product (as well as skin cancer deterrent) as it blocks damaging sun rays from your skin. SPF30 is advised as a minimum year round, reach for SPF50 in the summer, especially if acid toning. Reapply every two hours when you are in the sun for prolonged periods of time.
Suggested products: La Roche Posay Anthelios, Biore UV AQUA Rich Watery Essence
Google at your leisure I'm too lazy to link it all up
So in order of trialling when starting from scratch: Cleansers first (easy to see whether you can get on with them or not quite quickly within a few days). Then introduce moisturiser, then SPF, then serum, then contemplate acid toning. Aside from a cleanser I usually wait 2-3 weeks before introducing a new product.
Good luck!