I just bought an LS-300. I have no lenses for it. I want to start out with a simple lens, aka an inexpensive one.
I was thinking of the Panny 14-42, 35-100, 12-60, 14-140 or any other zoom lens you can recommend. I'd also like a good prime, but I'll start with a zoom.

Thoughts?

The Panasonic 12-35 f2.8 works very well on my LS-300. Stabilization works properly (controlled on the lens). All of the Panasonic and Olympus pro series lenses seem to get good reviews. Mostly I use full frame glass. That allows me to use the VSM digital zoom feature with both 4K and HD. Since I am a still photographer I have a lot of full frame primes and zooms available. My m4/3 Panasonic lens just covers 4K region of the sensor so I can't use the digital zoom feature unless I go to HD.

The Panasonic 12-35 f2.8 appears to be an expensive lens, at least to the ones I listed. Is there a cheaper alternative? After buying the camera, I need a budget lens, for now.

I think any of the Panasonic/Olympus M4/3 lenses will be fine with the limitations listed above. The attached is the latest JVC lens test results I could find.

Tagging onto what Luke posted, this is a list of 4/3rds lenses but un-tested with the LS300. Who knows, there may be something there that might be able to work enough.

The Complete Micro 4/3 Lens List (http://hazeghi.org/mft-lenses.html)

I have a couple nice early Nikon lenses and thought about using them with an adapter (didn't cost much) to get me by but they would have been fully manual.

If you can only afford one good all around lens, I would go with the Panasonic LUMIX G VARIO 14-140mm f/3.5-5.6 O.I.S. You can pick one up for around $500. It's not a fast lens, but in most other respects it works well with the LS-300. If you need a faster lens, it unfortunately will come with a higher price.

I picked up 5 or 6 Rokinon Cine primes. So far I use the 24mm Rokinon Cine the most. T1.5 and pretty good. I don't recommend getting the full set like I have.. don't need all the ones I have. 8mm, 16, 24, 35, 50 and 100 (Macro... ) plus a bunch of old Nikon MF lenses or if you get a Canon adapter you can get lots of old used manual focus Canon lenses. I think in general if you wanted fixed focals, some of the new Rokinons are better than the older Nikon or Canon manual focus lenses by a long shot.. but you can pick up a 20 year old Nikon or Canon MF lens for $50-150 vs $499 for new Rokinon. Just food for thought. IF you do want to put some old lenses on your camer (metabones Canon/Nikon to MFT adapter for $150ish.. pick one...) and look up lenses on KenRockwell.com: Photography, Cameras and Taking Better Pictures (http://www.kenrockwell.com) site. He has fairly in depth reviews and comparisons of lots of new and old Nikon and Canon lenses and some Rokinons. His thing is NOT videography but photography...., however his lens testing is fairly in depth and worth a look before buying random used lenses on ebay.

Also if you do go with new Rokinons, my gut feeling is it's better to get it in a Nikon or Canon mount instead of MFT so you can always use a speed booster 0.71 (Canon or Nikon booster from Metabones) to add to the lenses options. If you get the MFT obviously you can't use a speed booster. I haven't used a speed booster so I can't say if it's worth the $500 adapter.. but it's still an option.

I think any of the Panasonic/Olympus M4/3 lenses will be fine with the limitations listed above. The attached is the latest JVC lens test results I could find.
Recommended Lens List for the JVC GY-LS300 by JVC - file:///C:/Users/Aaron/Downloads/jvcjyls300_tested_lens_info.pdf

Thank you for the list. It is greatly appreciated.

I'd go for Nikon F mount lenses (as there are more of them, and are easier/cheaper to adapt than Canons) with the LS300.

Rokinon Cine Lenses – David Peterson (http://ironfilm.co.nz/rokinon-cine-lenses/)