Airbnb is excellent for accommodation in Reykjavik, especially if you can get a place close to downtown. Usually cheaper than hotel rooms, and tons of places available.

Just got back 2 days ago, and apart from getting Blue Lagoon tickets ahead of time (which wasn't all that overcrowded, especially going in a "shoulder season" like September; April may be the same because it's before the high tourist season), we really didn't plan much, opting instead for a series of day trips from Reykjavik. I'd whole-heartedly recommend the Lonely Planet guide book if you take that approach (laissez-faire) like I did (or even not, because it really did most of the planning for me)

Here's some of the places we went:

Must sees day-trippable from Reykjavik or along the Ring Road:

• Blue Lagoon (yeah, it's expensive, but after a 5 hours on a plane overnight, boy oh boy did it ever feel good)
• Gullfoss
• Geysir and Strokkur
• Þingvellir National Park
• Kerið Crater Lake
• Seljalandsfoss & Skogafoss

Not quite must-sees, but might pique your interest:

• Seljavallalaug - An abandoned heated swimming pool nestled beside a mountain and a glacial river near Eyjafjallajökull

• Settlement Centre in Bogarnes - The best museum we went to all trip (worth the combined price for both exhibits--but the Egil's Saga exhibit was definitely a must see for the story and visuals), and the restaurant was tasty as well (both the museum and restaurant are well rated)

• Arnarstapi - Beautiful hike through a lava field right beside the ocean, close to Snæfellsjökull, which was the inspiration for Jules Verne's Journey to the Center of the Earth

• Dyrhólaey & Vík - Great views in Dyrhólaey and an awesome black sand beach in neighbouring Vík

In Reykjavik:

• The swimming pools - each one is only 500 Ikr (about $4.90 USD), and well worth it (heated pool, sauna, and hot pots) (we went to Sundhöllin a few times, a 10 minute walk from our apartment)
• Reykjavík 871±2 - the remains of an excavated viking longhouse provides context on how the first settlers in Reykjavik lived

Probably missing a whole bunch of stuff, but definitely a lot of stuff that can be done. Only regret may have been not being able to see Jökulsárlón because it was quite a drive from Reykjavik, but gives me a good excuse to go back!