What can I do for better lighting in my rental galley kitchen?

I love the under cabinet led strips. They put out quite a bit of light and no heat.

You could add a string of LED lights under the cabinets inexpensively.... https://www.amazon.com/s/ref=nb_sb_ss_sc_4_18?url=search-alias%3Daps&field-keywords=under+cabinet+led+lights&sprefix=undercabinet+LED+l%2Caps%2C410&crid=21C9JYES99R3U

You could also put LED light bulbs in your ceiling fixture is its regular bulbs.

Dear Idalee, Try to purchase lights that are portable. You will only have to patch the small holes, when you move out from the apartment. The screws will only make small holes. Go to a Home Depot or Lowe's, and check out the lighting department. Also check a local hardware store, I'd call first. Good Luck

Check with your land lord before any improvement, but under the cabinet light strips like Carol suggested are amazing!