Paint cabinets as last resort. Even if you send doors out to be baked, they won't hold up to what u have now. Start small. Switch out accessories. Change out hardware. New chairs at island. Paint walls. New pendants, then perhaps if you are unhappy paint cabinets.

Why not wait to make the decision about painting after you've changed the backsplash? The cabinets look like they are in good condition. I go by the adage of "if it ain't broke, don't fix it"

Hexagon white is much better than what you have now. But hexagon to me belongs on a floor, I'd prefer to see subway tile.

I wouldn't paint those cabinets and just get a pretty, neutral backsplash. Like this color/size...

Cabinets are beautiful! Change the backsplash only.

I think that once you change the backsplash, you won't feel the need to change your cabinets. The current one is both too dark and too busy, it drags the whole kitchen down with it.

I love hexagon tile, but I will point out this- it is most commonly used for floors. And bathroom floors at that. So it might feel 'off' to you once installed just because of that unconscious association. If you're really sold on hexagons, I wouldn't go smaller than a two inch size. I would recommend an elongated hexagon, something like this. Here it is with a raised frame for added interest. and here's yet another option. You could even get a mix. here's what appears to be the same tile, for way cheaper. even the framed option and the concave one. You would need to decide what direction to install it, though I would recommend horizontally. Like any hexagon tile, it will be a little frustrating to install compared to just plain subway tile, so keep that in mind, whether you do it yourself or you hire someone to do it.

If you were to change anything about the cabinets, I would consider to something about the glass door. They're really best suited for cabinets with china or glassware in them, or are at least with contents that are fairly uniform. You could put the contents of everything in there into prettier containers, but not everyone has that kind of pinterest-perfect lifestyle where the have the time, energy or means to do that kinda crap. You could try to see if you could get something more frosted.

Changing the knobs with something slightly larger, in a brushed nickel finish might also make a difference.