Now that the days getting longer most operators stop going to hunt for the northern lights. That is due to the fact that we need darkness to see the northern lights. It could maybe be possible to see them during the few hours left of darkness now in Spring but the tours would start around midnight ending at 4 am which are not very popular times for tours.
This is however no reason not to come to Iceland. If anything maybe the opposite! In April and May we are between seasons and something one could call a low season. This means less visitors around the countryside which makes it possible to have a more real and authentic experience. While Gullfuss, Geysir and Þingvellir are always popular now is the time to more than the golden circle. The long days are perfect to explore regions like the Snæfellsnes peninsula. Daytours to the peninsula take about 13 hours so long hours of daylight are predestined to explore this gem. The peninsula is often called Iceland en miniature due to its many different landscapes. It is also the point chosen by Jules Verne to enter through the volcano Snæfellsjökull and initiate the journey to the center of the earth in the book of the same name. Along the south you can also start considering going further than the town of Vik and visit the glacial lagoon Jökulsárlón where Icebergs float into the ocean.
To cover more you can also consider going around the ring road, just beware that mountain roads are usually closed until Mid June. Their opening does not have a fixed date but depends on the conditions. So unless you have or hire a Superjeep you should wait with your visit to Landmannalaugar or Þórsmörk.
For the days between tours you can stay in Reykjavik and on sunny days see the locals sit outside in shorts and short sleeves even though the temperatures may be only around 5°C. With the short summer every sunbeam counts! Another fun activity in the city is the Zoo which is a cute collection of local animals such as cows, Icelandic horses and other animals found in Iceland. In spring you may be able to see some new born animals which specially for kids is a great experience.