1) Java: a good option is Batu Keras, a protected bay that is perfect for learning. Not too busy and quite a nice spot. http://www.javacovebeachhotel.com/thesurf.php
You can always hire a local and a board and take you out there.
2) Bali: dtc's advice is good. The only real learner's break is Kuta Beach, the other place where people learn is Echo Beach in Canggu which is a lot less hectic than Kuta. Any surf school is ok to teach you the basics. Some take you to other breaks too but all you really need is a beach break, a board and the basics.
3) Lombok: there are some surf school at Senggiggi Beach (tiny surf mostly) in Lombok and also in Kuta (Lombok, same name as Kuta in Bali, confusing I know).
It really depends where you want to be generally, as you will be only spending 2-3 hours per day in the water as a beginner. So pick the island where you want to be the most. Indonesians are enterprising people. Wherever you have surf, you have someone to rent you a board and teach you the basics.
I personally think the surf camps with accommodation included for beginners are really only for people that need hand holding and haven't got any travel experience, a waste of money. A surf school where you pay for the day is fully sufficient which also gives you the option to opt out if you don't like it and find another course you like more.
May is the start of the season. By June you will see some big swells rolling through. Take care and enjoy!