For Psoriasis: In most cases the dermatologist will just need to look at your skin with a magnifying glass (this is accurate in diagnosing psoriasis up to about 97% - my P is a little atypical and that's all it took for the doc to diagnose). Sometimes they need to do a biopsy, but this is generally if a couple different treatments don't work (usually they will try steroid topical, anti-fungals, and antibiotics - if none work they'll biopsy).
For PsA: Usually you'll have an x-ray of wherever your joint pain is the worst or has been present the longest. You'll also have blood work, usually to test CRP & ESR (these are inflammation markers - it is common for them to be normal unless a nasty flare is happening... Mine is usually 1-2 on a regular day and around 15 during a flare) and usually a couple antibodies (like Rheumatoid Factor and ANA to rule out Rheumatoid Arthritis and Lupus (which are mimics of each other)).

There are other possibilities including a KOH prep (which is usually just a swab) to check for fungal infection, musculoskeletal ultrasound for PsA inflammation (it's better than xray but a lot of people don't know how to perform it), and various other blood tests. Sometimes nerve conduction studies are done to look for neuropathy (as PsA can cause this and it's an alternative cause of pain) or checking for co-occurances (thyroid, blood sugar, liver and kidney, GI tract, ect...) but usually only if there's something that suggests they should check that out.

I hope your dermatologist is more helpful than mine was. I was sent to her for what I thought was a weird rash on the back of my hand. She looked at it and said "you have psoriasis" gave me a cream and sent me on my way. I saw her one more time and questioned PSA, she said "oh no, it's not that". Now 3 years later I've seen 2 rheumatologist, the 2nd of whom was with me less than 10 min and diagnosed PSA, with a likelihood of fibromyalgia. The best advice I can give is like my best friend tells me all the time "you need to be your best advocate, you know when something isn't right with your body."

Sorry, it took so long to diagnose. My first plaque was on my temple after a half marathon where I had bobby pinned a headband. My GP tested for fungus or metal allergy, but it was not. About 4 years later when I got another spot on my scalp and my new GP who is an internist told me it was psoriasis. I went to a dermatologist who upped the ante and diagnosed me with Psa, but not before I'd been to a surgeon to get my knee (pain) x-rayed for internal damage (nothing showed). He guessed it was likely a torn meniscus, and started me on therapy. I quit therapy because it seemed pointless. My derma diagnosed Psa primarily from the symptoms I told her and a physical examination. I am currently mostly pain free and grateful. I think the Olive Leaf supplements and Neem Leaf helped.

I have had psoriasis for years. On a visit to my 1st Derm she looked at my red palms and fingernails which are lined and said I had psoriatic arthritis. She was right. I made the mistake last Dec and March and got 2 injections of Kenalog. I won’t do that again. My hair is falling out, I have atrophy and tremors. All with the flaking, itching, on my elbows, arms, face, head. Not what this 69 year old wants at this point of my life.

I just had a biopsy done on mine, I still have the stitches😯 it's not that bad, I have a great dermatologist😎 I wish you luck.