April to July will be damn hot in Goa and then Aug to Oct, beach activities goes off for security reasons.
I would suggest... you should plan Goa trip in Dec-Jan time. So you get to experience Goa as the best.
See goa in june is good if u love monsoon, if u wanna go and just relax and enjoy the rain then it's good option. People visit goa to enjoy the beach life but during that time because of high tides or heavy rainfalls you may not able to do that. You can visit dudhsagar waterfalls, best option to visit during monsoons.
Firstly goa will not fail to surprise you be it any month or season of the year. Secondly, it depends on your taste and preferences because June will be mostly hot and mix of monsoon season. Also, there will be less tourist commercialization than other months due to off season, and also due to off season and high tides some of the beaches might be closed. So think accordingly before planning.