I would do can (recessed) lights in several spots and remove the ceiling fan and rectangle lights. It looks like it is not necessarily an eat in kitchen area (judging by where the table is located)?? If not, I would leave it at that, otherwise, I would place a drop pendant over the center of the table.

Consider the lighting you'll need. Task lighting with under counter lights. A pendant above sink and can lights around room to warm and brighten it. If you want a separate space for the table in the kitchen, forgo the pendant above sink or make it a simple light and ad a nice fixture above dining table to note that area. Hope that makes sense.

Agree with recessed lighting. A fan isn't a bad idea over dining area either. I could see up to 6 can lights in your room- though only 3 would probably work fine. A lot of heat can be lost if these lights aren't installed correctly so be sure of the installation. Get IC (insulation contact) type cans and lens.
Consider where you would like the wall switches for your lights. Its good to have the lights on different switches because usually you will only need minimal lighting. You might have some lights on 3-way switches as well,

Fluorescent tube needs to go. Tear it out first for the sheer satisfaction, like pulling a nasty splinter.

Include under cabinet lights in your design, above all of the counter areas, and think about whether you want them on a wall switch or individual switches at task locations (or both).

Avoid fluorescent lighting of any kind. Go with LED instead, and use incandescent or halogen in the range hood.