Since you only have a week to spend in Indonesia I wouldn't recommend staying in Jakarta. It is a great city but you would really need more time to experience both Jakarta and Bali/Lombok. Also Jakarta isn't too romantic....you would probably end up spending the majority of your time in traffic there. But if your trying to paint, stay in Jakarta for a bit.
You can fly one of the domestic airlines to Bali for fairly cheap. I'm sure your budget is pretty stretched with so many destinations, so what kind of accommodations are you looking for? Are you just looking for just an air conditioned room? Or something more luxuries? For budget accommodation I would recommend Masa Inn on Jalan Poppies Lane #2 (in Legian/Kuta area), but there are plenty of budget hotels around the area. Like I said before....take trips out side of Kuta because it ends up being kind of a tourist trap at times. Check out Ubud (monkey forests, local artists), local dances/temples, Jimbaran Bay (great fresh seafood diners on the beach), Dream Land beach to name a few. Its worth renting a little motor bike or car for a little freedom to get around so you don't always have to take taxi or walk. You might want to look into getting and International license since your going to so many places to keep things legal. But you can get by renting the bikes with false info. so its no big deal. Just be prepared to pay a little when/if cops stop you for a license check.
As for Lombok, you can take the ferry over (about 4-5 hours) or fly (30min) from Bali. Either way is pretty easy but the ferry is going to be pretty ghetto, but a good experience for sure. Check out Kuta (shares the same name as the area in Bali) or Gili islands (might require another boat ride). I haven't been to Lombok since I was real young so I can't speak on it too much, but it is a less crowded with tourists than Bali.
Also, you can purchase VISA on arrival for US$20 or $25, that makes it very easy and when you leave the country you have to pay a minimal exit tax. Look into this stuff online to get the specifics.
