Congratulations on your marriage. Italy is such a lovely place for a honeymoon.
It is Jubilee year in Rome and there are also some holidays to be aware of, 25 April is Liberation Day and 1 May Labour Day. Just something for you to be aware of in factoring in your planning.
Without understanding your budget, for Rome, I would choose a centrally located area. We particularly like either the area around the Parthenon (which is convenient to many of the Tourist sites) or the Trastevere area which a bit more residential, but still easy to get to most of the tourist attractions.
For the Amalfi Coast, in early May, you have a couple of options. Transport on the Amalfi Coast is a bit challenging ( you don’t want to drive) and the buses can get crowded. If you really want to go to Capri, Naples, and Pompeii, then Sorrento is a more central transport location. Not a beach town, but great transport options, food is amazing. For the Amalfi Coast, we have stayed in Positano before and loved the B&B we stayed in, however, it is very hilly and transport is a bit more challenging and it is very expensive. Amalfi is a bit flatter and a better transport hub, but not great beach options. Since the beaches on the AC are not your traditional sandy beaches, if a beach is important to you, then look for hotels with a private beach option.
Note that when you plan, weather can be a factor on the Amalfi Coast. We were there last May (end of May). It rained most of the time, ferries were cancelled and our beach vacation morphed into something else. However, even in the rain Sorrento and the Amalfi Coast was magical. They have wonderful wineries, local food tours, cooking classes. We will be going back as we still want our beach vacation.
You will get some amazing advice here on the forum. As your plans continue to evolve, happy to help with any specific recommendations.