Sorry in advance for the essay...
We went at the end of October and it was perfect. It was chilly without being intolerable, but beautiful sun and clear skies. We were lucky though, it isn't generally that consistently clear.
We quite liked the Blue Lagoon (the face masks were great for my skin at least, and the restaurant is absolutely lovely) but we made a full afternoon/evening of it with dinner after, and it wasn't particularly that busy. It is a lot more touristy (they have staff who will take your photo etc) and there's sludge that builds up in the rock crevices which can get a bit whiffy.
We also did the Sky Lagoon which was much more beautiful and only 15mins or so from Reykjavik. It's an extremely lightly chlorinated pool (you can't actually tell - no stinging eyes!) that overlooks the sea, has a little waterfall, and is simply breathtaking. The sauna has the best view of any sauna I've been in too. It was really windy and started raining when we went but we didn't mind at all because it was so lovely. Of the two I'd choose that one
The northern lights tours usually let you rebook so it's worth trying early so you can go again later in your trip, but it's a long miserable night if you don't see anything. Keep an eye out on the northern lights trackers online so you aren't booking when the conditions are less than ideal and wasting your time. We were lucky enough to book when there was a solar storm so they were dancing in glorious colour, and it was honestly one of the most memorable nights of my life!
We did day tours of the West and South coasts and the Golden Circle - they're long days but worth it. Next time we go we'll be hiring a car, but it was worth having a structured tour the first time around so we could ask questions/get timings right/learn a bit of history. Have a good think about what you want to see most (geysers/black sand beaches/waterfalls/glacier etc) because that will dictate which tour you end up doing.
We pre-booked all of our tours and restaurants before we went so we were guaranteed to get spots. It makes the trip itself much easier.
Either way, it's an absolutely stunning place. Definitely budget a lot for food and drinks though because it isn't cheap, but the quality is worth it, even in the little cafes and breakfast spots. We had some of the best food we've ever eaten!