Hi there,
Few more things. For a weekend you can drive about 2-3 hours to West Java. Things that you can expect to see are as follow (in distance-order):
1. Bogor-an old Colonial like city. Kebun Raya Bogor (rare plants, huge trees, etc).
2. Puncak-mountainous district, tea plantation, Puncak Pass (top of the mountain), Pangrango Mountain. Beware of jam, this place is famous for the jam.
3. Lido Lake - it is a lake, down the road is Bodogol Jungle Park (?) Forgot the exact name, it is tropical rainforest area (much much better than Bukit Timah).
4. Gunung Halimun National Park - it is indeed a huge real tropical rainforest with few waterfall, and awesome clear river.
5. Situgunung Lake - A vast lake with a beautiful waterfall.
That's more for nature side. All of them are located south of Jakarta.
In Jakarta itself, few more interesting locations is at the central of Jakarta. Monas (National Monument), Cathedral and Istiqlal Moswue (biggest in ASEAN), you can go to Ancol-North Jakarta seaside (there's an old Dutch/British Tomb at the side with a fishing pod made of bamboo), if you kinda like antique you can go to Kemang lots of antique shops there and some nice houses, etc.
Anyway, since you are in Jakarta for 3 months. If you have the time, I would like to recommend you to go off Jakarta to other cities.
- Jogjakarta is 1 hour flight or overnight train - this city is amazing. Borobudur Temple, Prambanan Temple, Keraton Palace, Merapi Mountain, beaches, etc. I did go around Jogja for 2 days. I can give you my itinerary if you want.
- Bandung is 2 hours drive from Jakarta. Another old cities with Tangkuban Perahu Crater, Kawah Putih Crater, many colonial houses, etc.
If you like adventure and do have time and guts to travel to non conventional places, Java Island has lots of it. I may be able to help you once I'm back from my backpack trip to Sumbawa in early June. Do PM me if you need more details. Cheers!