I just started composting. In addition to putting lawn clippings, kitchen waste and dry leaves what else can I put in? Is it okay to put branches of trees (fruit, olive, etc) in the compost?
Dry leaves should be shredded first as otherwise they might form an impenetrable mat in your compost pile. If you don't have a shredder, it's easiest to just collect a pile of leaves and run the lawn mower over it a few times. This provides the "browns" or high carbon material for your pile.
Composting uses bacteria to break down the organic material. But wood needs fungi, a different microbe that breaks down lignin at a different rate from non-lignaceous products. So, you'll likely end up with all the branches largely intact.
It's better to just bury all your woody material and plant on top of it.
I used to put branches on my compost pile, but it takes years until they decompose.
However it is possible to use them if you reduce them in small pieces with a shredder (if not too many of them). If you usually put your mowed lawn on it, it will be very good to balance between nitrate rich and carbon rich inputs...
The humanure manual provides interesting information about how to handle a composting pile - even if its not to compost humane manure.
About the shredder, which can be expensive depending on the size of the branches you have, you could see with your neighbors if you can borrow one, or by one together with them.