Bracken Joy, I second the recommendation to read through the Starting from Scratch wardrobe series on the Vivienne Files. It's a bit of a rabbit hole, to be sure, but I learned a ton and am moving forward with confidence. The first thing that helped me was to think about--as others have already said--2 basic neutral colours that you prefer, and 2 accent colours. I looked at my current wardrobe and it became clear that charcoal and black would be my neutrals (oh the relief from considering whether I should buy camels, navy's, etc!). You also then add denim as its own category of neutral, which is also super-helpful as dressy denim is a big part of what's available for business-casual wear right now.
So charcoal and black for you would work: at least one of each in a pair of pants, plus a pair of dressier denim. Then, personally, I'm not comfortable in blazers, so I'm in cardigans all the way. These should be more forgiving with your arm length, as they can be worn at 3/4 (pushed up a bit) or in a looser style that the flowy shells have. One of each in black and charcoal, if that's what you're going with.
Then you have 3 colours for tops: whatever shade of white works on you (I can't wear pure white--too stark), and 2 accent colours. From your photo, I'd suggest green and red, but you could also go with whatever colours you prefer or like to wear most often. Again, I found it so freeing to just focus on every shade of blue and purple--from baby blue to navy, from lavender to indigo/violet--and to let go of all the others. In your case, you'd have every shade of green (for different seasons too) and red. Some of your current t-shirts could definitely work, and you can build from there.
The website also talks about thinking about your preferences for oxford, button-up shirts vs flowy, drapey blouses or tunics, or pattern, or whatever. Just recognizing that you should acknowledge what you feel comfortable in and stick to it; stop picking out things you think you *should* like, but that then you never really wear. What a thought! :)
So you would end up with an easy wardrobe and daily routine of a pair of pants, a comfortable/professional shirt, + a cardigan or whatever you decide you like as a shell. Every colour of each will work in an easy rotation, and you build accessories to change things up a bit more. I am not really a jewellry person, but I love scarves, so I have collected a few in different, plain colours (I'm a colour-blocker, not really a pattern person). Wouldn't that be easy? :-)
I wanted to add, too, that I'm working in that same west coast, laid-back, business-casual world. Socks are only an issue for you to decide around comfort. Some women like blazers, but there are no suits. No heels, although some women like that 1" that Mrs Peacock was talking about. Some women wear only pants, some are more comfortable in dresses. Totally up to you.
Personally, my uniforms for winter are a pair of blundstones (which I wear with wool socks, but I've done lots of Clark's shoes over the years and haven't done ballet flats precisely because I LOVE my wool socks in winter) and some not-too skinny charcoal pants, or boots and skinny dark denim, plus a camisole that I can tuck in for warmth, and then a merino wool sweater (I bought 3 this year from Costco! They had all my colours: black, purple and light blue!) and a more decorative scarf or sometimes a glass pendant. I am still looking for some lighter blouses and need to add a black cardigan. I have some lovely black trousers, but need some black shoes to go with them. Again, the Vivienne Files pages were so helpful for me to assess what I already had, and what gaps would be useful to fill.
I am also considering adding a dress or too: nothing to formal, but something I could wear either with tights or black leggings, and either my blunnies or my boots. I think that would be warm and comfortable for winter.
In thinking about your situation, I also think if I was going into people's homes, I wouldn't want to dress too formally. I would want to be professional, clean and tidy, but not so formal as to be dressed way above the way people dress in their own homes, as I would want to be able to put them at ease. I would also probably go with socks, as here at least most people take their shoes off when they enter someone's home, and I'm not really comfortable barefoot in that scenario for some reason! :) But YMMV on that one!
(is it too unmustachian to admit that I'm going shopping this weekend?! :) Honestly, it's a wonderful feeling to know what I need and know that when I find it I will wear it for years to come.)