I see an adorable small kitchen. On shelf under kitchen sink, store your supplies in natural wicker baskets.

In a small kitchen, keep counters tidy. Store liquid soap etc in stainless steel containers

What is the flooring? Wood? Do you have a rug on floor?

Paint walls white and buy a colorful painting for the eating nook wall.

If you are like me who cannot tackle the remodel on one big project expense, I would start with your breakfast table nook. Then I suggest working on the cabinets, and lastly changing the tile wall with or without counter top/sink update. An update of the faucet at the start won't be too much expense either.

I am assuming you're not fond of the blue tile - wanting it to be in off/ white or grey tiles eventually. If you absolutely cannot live with the blue tile while progressing with the project, I suggest painting it like several commentators suggested as a temporary fix - just to get the pain out from looking at it :{)
Otherwise, I suggest to leave it alone and save your money, time and effort.

For the breakfast nook, I think additional storage is always an advantage like two commentators suggested for your wall by your table - using clean line metal/steel brackets and white shelves and wall painted with light cool grey in your ideabook. You can display not only kitchen items but also some artwork on the shelves to make it look uncluttered. I also like your posted ideabook for a bar style chair and table (Mal Corboy) but with low open back chairs for airy look (Kelley Gardner.) Assuming it is safe to cover your radiator panel (sic maybe - not sure of term), the wall end of table can be opened with cabinet doors on both sides or at least the one facing the boiler side. (Pls refer to 1st image.)

For the cabinets, I suggest going up to the ceiling even above the window/sink - using simple panel doors with long sleek hardware just like Merry Powell suggested. Going up means less wall tile to change later in the project - an added advantage. I also think that painting the lower cabinet that same light grey as your wall also looks pretty just like in your Cucinabella Bella choice in your ideabook while top cabinets are in white. Just be careful which white shade you choose. A true off-white may not look good with the grey you posted in your ideabook. Your boiler corner can have its own two-door corner cabinet or 4-door cabinet if you paint the lower ones grey. (Pls refer to 2nd image).

I am attaching images with edits using my unglamorous graphic 'editor' - excuse the roughness ;) I also included images posted in your ideabook mixed with my added images. May be a lot of work but have fun while doing it. I'm sure it will be much better looking each step you take.