If you are looking for options to stay, I'd suggest that you stay in either Mumbai or Pune because they are nicely connected with every travel destination of Maharashtra. If you are looking for places to visit around Mumbai, you can opt for Mahabaleshwar, Lonavala and Lavassa.
In and around Mumbai in May will be too sunny and hot. I will suggest you to visit the hill stations nearby to cheat the heat and enjoy the pleasant weather at early morning and night.
You can visit: Igatpuri, Saputara, Lonavala, Mahabaleshwar, Matheran.
U can even visit Alibaug.I will recommend a ferry ride from Mumbai to Alibaug.
Or if you are a wine lover visit Sula Vineyards which will be a fun experience for you