Given your limited time in Indonesia, I would advise you to skip Medan and Lake Toba (I was rather uninspired by Lake Toba when I was there earlier this year). Papua will be very difficult logistically in your limited time because you will need police permits which take time to process. As
lingua101 suggested, I would also advise you to just enjoy the stretch between Jogja and Gili.
Jogjakarta
2 full days in Jogja should be more than sufficient for you -
- Day 1 : Visit Prambhanan (you get free bus to Ratu Boko with its nice view after that). Get back to town and explore around Maliaboro Street (Kraton at the end is very unimpressive so don't waste time there) and Taman Sari (most of these can be done with very cheap public transport)
- Day 2 : Hire a private car for the full day - ask to be brought to Borobodor (go early to avoid the crowds - or super early to catch the sunrise). Enroute back, stop at the delightful Ullen Sentalu museum which will give you a good insight into some Javanese culture)
You can then move on to catch Mt Bromo & Kawah Ijen
http://www.bemytravelmuse.com/mount-...ithout-a-tour/
I haven't been to the Gili Islands so can't advise. Likewise for Lombok which is supposed to be the new Bali.
But for Bali, yes - head straight to Ubud and use that as your base. There will be A LOT of tourists in Ubud but it is still a much better place than Kuta/Seminyak. You can do a host of activites there to "immerse yourself into the environment/culture" eg :-
- A bike ride through the padi fields (it's easy given they bring you by car to the top of a mountain and you just mainly go downhill 90% of the way)
- Taking a morning/evening stroll through the Campuhan Ridge walk and enjoy the perfect view at the Karsa cafe at the top
- Go for an early morning hike to catch the sunrise at Gunung Agong
- Catch an evening performance of the kecak fire dance
Hope the above helps!