Swedish word of the day: Aprilväder (April weather), a varied and unpredictable weather common in April.
So basically bring everything :) Joking aside, layers is the key. Expect temperatures from around 0 or so to 20 in the day time, the evenings and nights can be cold with temperatures dropping to 5-10 below zero, but can also be a lot milder. And don't be surpised if it snows. You are staying in southern Scandinavia so it will not be too cold (in the north, the ski resorts will still be open), but higher up (like on the NiN trip) it will be bit colder and can be snowy. Bergen can also be very rainy. So, I would leave the long underwear and scarf home. As for jackets, I use a thin wind- and waterproof jacket this time of the year and then add layers underneath if needed.
As for day trips from Tallinn, Tartu is a popular choice. A bit more odd is Narva on the russian border, most of the medieval old town was destroyed by the Soviets in the 2nd world war, but the 13th century castle is still standing. As well as the fortress on the other side of the river. Visiting the fortress will require a russian visa, but it can be admired from the Estonian side of the river.