Bring a lightweight rain coat just in case, maybe a scarf but I don't recall ever needing anything else besides a long-sleeved T even while biking. One thing that does happen across Sweden in September is some hours are reduced at attractions like manor houses, so check opening hours for your time period. The more famous places may well be open all week through October and almost everything is open on the weekends.
- And since you mentioned biking.... I LOVED biking in Copenhagen! It's a treat, great signage and almost everyone actually follows the rules. Unfortunately, the last time I biked in Stockholm a few years ago, there were those small electric scooters driven without regard for others on bike paths & it was not enjoyable.
We like poking around cute towns, historical corners of older cities, unique local sites. Not especially big on castles or churches but historical walks
In which case, Gotland & Visby would be an absolute treat. We stayed at the Best Western in Visby & rented a car for May 1st holiday weekend & it felt like there were very few tourists around. The town is very walkable, beautifully preserved medieval wall around it, plenty of cathedral ruins.... you get the idea.
If you’re planning on day-tripping or overnight outside of Stockholm, I’m biased but Uppsala is a fantastic university town. (Think Palo Alto but with an 600+ year old university & cathedral). I've had a number of visitors who ended up staying in Uppsala & taking the train into the more expensive Stockholm rather than the other way around. https://community.ricksteves.com/travel-forum/sweden/uppsala-where-to-stay
I wrote up my favorite spot in Stockholm for spending a day here. https://community.ricksteves.com/travel-forum/sweden/4-days-in-stockholm-a-day-trip-worth-it.