I understand wanting to change the backsplash. It would probably require removing the old - and doing a little wall repair. Usually, it would not be run on the wall with the window like the present one does but stop at the corner.

We usually recommend that cabinets be painted by a professional if you are not an experienced and patient DIY person, or are only trying to extend the life of your cabinets a few years till you get new ones. Cabinets are a big investment and a bad paint job can end up costing you $. Also, whether DIY or professional, the finish will not be as durable as a factory finish. If the only people using the kitchen are responsible adults - not a big deal, but hockey stick yielding children - might be a problem:)

I'd replace the backsplash and repaint the walls. The cabinets look very nice. Painting them in my opinion would ruin them.

In case you wish to change your kitchen cabinets, there are low-cost cabinets out there that are well-constructed, aesthetically pleasing, and exceptionally reasonable. If you decide to give your cabinets a new paint, make sure to consider a professional for this job. Painting cabinets only sounds like an easy and quick solution. However, if you want it well done, a professional is your solution. A glass backsplash sounds like a great idea. However, if your mind is set on renovating the kitchen, take into consideration all the characteristics of the marble countertop that came with the apartment - this material scratches, chips, and stains more than other countertop materials. If you like countertops that look a bit scratched up and that develop a patina of use over the years, then keeping this countertop is a great idea. If you want your countertops to stay glossy, quartz or granite would be a better idea.

Since you don't want to do too much to the kitchen, I would prioritize the tile as the first thing to do. You might want to consider tiling the whole window wall as well. It's a pretty small area (from the photo) and it could be a nice focal point when you enter the kitchen. I'd keep the pattern and style simple. Instead of the cabinets - I agree that you should redo them professionally, which will be expensive no matter what you do - I would consider replacing the flooring which is also dark. Again, maybe a tile that compliments the backsplash and wall. I'm not sure what the kitchen connects to if there is a natural break for the flooring or not. Good luck!