Yes, i would paint them. i would also remove the trim above the sink and replace it with something that has a cleaner line. They would look really nice with an antique type finish. A blue or soft green could be really pretty and make the tile pop. Lightly sand them and then once painted gently sand off some of the paint to expose some of the wood underneath. I think you would have to seal them with a finish afterwards. Then paint the walls a soft white.

congrats on your new home!

are the cabinets still structurally good? if yes, then yes I would paint them. I would take down the gingerbread over the window, paint the cabinets, update the hardware -- and if you can squeeze it into the budget -- replace the counter with a cherry butcher block from lumber liquidators (for the bargain price of $299 for an 8' length -- and sometimes it is put on sale for $100 off).

the absolute best paint that I have ever used for painting furniture is Benjamin Moore "Advanced" -- it pretty much eliminates all the finish problems that normally happen. but it is quite a bit more expensive then the cheap paint you can get at one of the big box stores.

There is also specific kits to paint cabinets. I think from Rusoleum? But I've seen them at home depot and mentioned here on Houzz. I've never used it, so I can't say anything to it's quality but it's worth looking into.

I have painted many cabinets, it isn't hard and the results are refreshing. New hardware, new counter top and you are good to go. I agree, you should do away with the upper trim around the sink. Rustoleum has great products that are strong and wear well.

You could do a small trim on them to give them some depth. I agree with removing the trim above the sink and painting them. Your floors look pretty dark so I would go with an ivory or white color! :)

What are the aspects of the kitchen that you like and dislike? Painting cabinets is a start, but if you really dislike some other parts of the kitchen more, it might be better to start with redoing those instead.