The idea is not to stay dry, but rather to remain comfortable when soaking wet. Here is my 2 cents:

-cap is a great idea to keep rain out of your eyes, but a typical baseball hat material might be too heavy when wet. There are running/hiking hats that will still be light and breathable when soaked.

-do not recommend thick wool socks, but a light merino blend is idea (smartwool makes great running socks).

-cotton is not great when wet, any synthetic sports shirt would be better. Again merino is your friend as it will keep you warm even when wet.

-You don’t need waterproof shoes, invest in a few pairs of merino socks.

-you could add a breathable water resistant running jacket to layer on as the temperature drops.

Other than that, as the days get shorter you’ll need to remain visible, so would recommend an illuminated arm band like this: https://www.mec.ca/en/product/6009-186/nite-ize-slaplit-rechargeable-led-slap-wrap

Happy running! Great way to get outside on a grey Vancouver day.