If I were going to this area, where there is so much to do and see, I would not be trying to cover the Azores too, as this is a 2.25 hour flight from Lisbon, so by the time you get to the airport, allow 2 hours before the flight then time at the other airport, you will have lost more than half a day in each direction. Your choice if you think this is worth it.
Flights to Porto Delgada are with TAP, Sata and the discounter Ryanair. Ryanair will be the cheapest, but be careful as they add on for EVERYTHING i.e. each checked in bag, food, paying by credit card etc. I always use them from the UK as a last resort.
Car hire is recommended once you get to the Azores so that you can see as much as you can in your time there. In April, the Azores will still be cool and wet and the flowers start blooming from May onwards, but if you are going at the end of April, you may get lucky.
If you wanted a long day trip out of Lisbon, take the train to Porto and try a river cruise and port tasting. The train would be easier than driving for Porto, but it is also worth driving to picturesque Obidos from Lisbon and picking up the coast road on the way back.