I plan to go from Senggigi (west Lombok) to Labuhan Lombok (east Lombok).
Is there any buses or shared taxi doing this journey?
If yes, where can I take it? how much is it? and how long does the journey take?
Get a bemo from Senggigi to Mataram's Mandalika bus terminal (Terminal Bis Mandalika), from where there should be semi-regular onward bemos to Labuhan Lombok (about 2 hrs). Prices are fixed and should be listed on the notice board at Mandalika, it was Rp. 11,000 a few years back.
The info below is from 2015.
In Indonesian: Terminal Bis Mandalika.
Take a Bemo:
I wrote a dedicated question for this: Mataram to Labuhan Lombok, how to?