There’s no such thing as bad weather, just bad clothing. As long as you dress appropriately and take notice of how quickly the weather can change in Iceland, you should be fine.
The key to dressing for Iceland’s climate is layering, regardless of the season. Lightweight woolens, rain- and windproof jacket and trousers, as well as good walking/hiking shoes are essential if you plan to do any touring in nature.
The fact is, Iceland is not as cold as one might think during the winter. We have on average temperature just below 0°C during the darkest winter months, but you might find that the cold here is a bit different from what you are used to. That is probably due to wind, and the way wind seems to amplify and exaggerate the cold.
When traveling to Iceland in winter you should also plan to bring a warm overcoat, hat, scarf, gloves/mittens, socks (preferably made of wool) as well as a sweater and/or cardigan, and waterproof shoes with a good tread. Long thermal underwear (a shirt and pants) and a layer of fleece on top are also recommended if you plan on spending lots of time outdoors. Crampons are not always necessary, but they can add a bit of security and confidence when walking on icey areas.
Summers in Iceland are not as warm as in the Scandinavian countries. We have around 10-15°C on average and nights, especially in the highlands and the north, can get a bit chilly in August and September. Then there’s the rain, we don’t often have showers of rain, but instead we often see days with steady, persistent drizzle.
In summer, carry a light, and preferably water-resistant, jacket.
As trails are not common, sturdy walking shoes for trekking & hiking are needed. Elsewhere comfortable sneakers and boots work well. As it tends to rain a lot, rubber boots are also a convenient addition to your Iceland wardrobe.
When out and about in the city or dining in restaurants, it’s good to bring smart casual clothes for going out.
And lastly, always bring a bathing suit! Icelanders’ favorite pastime year-round is outdoor swimming in the countless geothermally-heated pools and lagoons. You don’t want to miss out.