Scotland is known for constant rain and most people living here have adapted to the weather hardily. Since it rains most days here, we've had no choice but to accept the weather and move on. Not having a car also changes our views on the weather. Since we're not spending our money on a car, it makes sense to instead buy reliable clothing that will hold up to the bad weather, since we'll be walking nearly everywhere in the rain. I am a firm believer that waterproof, dry clothing is worth it's weight in gold. (Additionally, if you're going to visit Scotland, here is what I would recommend you bring!)
A sturdy rain jacket (preferably with a hood.) When we first moved here, the first purchase I made was for this warm, rain jacket. It's expensive, but I've worn it every day since we've moved here and it has held up amazingly well. Umbrellas are useless here (blustery wind and sideways buckets of rain) but a good coat with a hood will keep one dry from head to, well, waist or so. (I found Aaron a Barbour jacket 75% off and while that coat doesn't have a hood, he wears lots of hats to keep his head dry. Besides, he doesn't really have to worry about his hair frizzing in the rain and he says he's fine with it...)
I wish I had recommendations for waterproof trousers. I've joked that if I could make waterproof jeans, we'd be rich because jeans soak up the water and never, ever dry. Alas, we just have to make do with damp legs for a bit.
I would definitely recommend solid, waterproof footwear to keep one dry. I see so many college students and teens walking through cold, murky puddles in inappropriate footwear and I just cringe. Been there, done that, never again. On semi-damp days I wear my ankle booties (although they're just about to fall apart so I'm searching for a replacement- Clarks make awesome Gore-Tex Boots which would definitely fit the bill!) and on the very rainy days, I wear my Bean Boots, which I got on the off season at the outlet store while on our honeymoon. Second best investment ever because every winter since 2011, Bean Boots have been sold out and the waiting list has been miles long.
Of course, we wear hats and beanies, scarves, and gloves to keep us warm. They keep those cold, pesky raindrops away! I definitely have to say that appropriate clothing helps us enjoy Scotland so much more and now we don't have any excuse about the weather keeping us indoors.