4 weeks in Bali and Lombok

Hello. My partner and I are going to be in Indonesia for a month in August. We are flying into Jakarta but plan to spend our time in Bali and Lombok, including Gili Trawangan and/or Gili Air. I would really appreciate any help or feedback on the following questions to help our planning along. We only have a month away, unfortunately, so really want to make the most of it but have no interest to rush around trying to tick of all the tourist highlights. We would like to see some of the cultural highlights and some places off the beaten track and to spend about 2 weeks on some beaches at the end- we would like some quite chilled places with a bit of fun in the evenings if we fancy- we were thinking maybe the Gili islands for this?

1) How should we get to Bali from Jakarta- plane, bus-boat, or train-bus-boat? We thought it would be nice to do overland (and sea!) to watch the world go by, what do people think and roughly how long would it take?
2) Where would people recomend to go in Bali in two weeks so that it's not too crammed- we only know for sure that we want to go to Ubud.
3) Can anyone recommend any particular places to stay in Gili Trawangan or Gili Air? We would like a budget (but probably higher end budget) bungalow on/near the beach with fan, own bathroom, nice friendly owners and a good restaurant/ bar to chat to others and generally relax. Do we need to book accomodation here in August and if so how do we go about doing that for the budget places that don't have websites?
4) What other places would people recommend visiting in Lombok?

Thanks for any advice!
Rona