Choose Southeast Asia instead of Europe or the Americas. This will fit your budget, travel time and desire for an offbeat experience desire. May–June is peak summer, and no matter where you go you will find tourists, and hot and humid weather.
I’d suggest Cambodia or Vietnam. Both offer amazing experiences. While there are no direct flights to either from Mumbai, you will find budget airline options, with a halt in Thailand, Malaysia or Singapore, to Phnom Penh or Siem Reap in Cambodia, and Hanoi or Ho Chi Minh City in Vietnam. Both countries offer visa on arrival but remember to get a visa letter before you arrive in Vietnam. Hotels are from Rs 1,500–3,000 per night with basic facilities. US dollars are widely used in both countries.
If Vietnam is on your mind, then start with Hanoi. The city has a great vibe, delicious street food and interesting history. Take a three-day tour of Halong Bay to experience the natural beauty of the region. If budget allows, book a private junk boat for the tour— it will be worth it. Then, spend two days by the beach at Da Nang before hopping on to the Unesco World Heritage Site Hoi An to see the beautiful lanterns. I’d suggest staying away from the touristy part of town. Also, don’t forget to try the banh mi at the shop made famous by the late Anthony Bourdain. Next, fly to Ho Chi Minh City and spend the remaining days in this vibrant place. Visit the Independence Palace, the maze of wartime tunnels, and the local markets for souvenirs before flying back to Mumbai.