Congratulations on your new home! It looks clean and functional which is a good start.
You are on a limited budget which is good because I would not invest too much in this kitchen.
Painting the walls will give you the greatest impact. I can't really tell the color of the floor, but if you would consider replacing it, that would be a good jumping off point for color.
Gel Stain. Many here have used it on their oak cabinets with great success. But, you have to like dark wood. There are other products to revitalize wood, but I am not sure of their success.
Hardware. I personally don't like the no hardware look.
Throw rug near the sink for comfort and color.
Since we don't know your budget, I am going to put out what I would like to see.
If you are planning to keep these cabinets for a good few more years: New counter, backsplash, new floor. All of these items have a huge difference in price depending on what you want to spend.
The other option is to leave as is(but just paint the walls) and concentrate on new furniture for the living room.

you can tweak and such but there's not much to this set up except some basic wall storage for dishes. No fridge/old dishwasher/hood light that likely does not vent/tiniest of silverware drawer.....Isn't there another wall of cabs? don't do anything until you're in for a bit. You need appliances and that costs and you'll come up with a better plan so they would get shifted around. Find out if there is venting over the range-you could probably get a scratch and dent or used over the range microwave... and have a full functioning unit,not sitting on the counter for at least initially...[the mwave will let you prepare things quicker so the marginal nature of the space won't bug you so much for at least a while and you can plan.]

Congrats! A first home is very exciting. You probably have lots of things to buy and that means lots of expenses. Don't fret about the kitchen, it is perfectly serviceable as long as the sink works and the stove works. If the dishwasher doesnt work it just becomes a great big drying rack-LOL.
My first kitchen was a real hole, 2 burner hot plate and running cold water only! Lived in that apartment for 3 years!
Some color would be great. Paint the kitchen, get a colorful throw rug, hang something in the space between the stove and hood. For extra storage/counter space, you might look on Craigslist for a kitchen cart, freestanding island, or bakers rack. Don't know what the other side of the kitchen looks like, is there a space for a table? Then you might need a pretty pendant light.

Re paint-Don't do it unless it needs a coat of paint. The first thing I did upon moving into the house is paint the kitchen. Since then I've painted practically every room in the house...but....the kitchen is one of the hardest rooms to paint/you'll want higher quality paint to make it worthwhile and technique is established only after some experience painting as there is so much brush work with cutting in and edges....the living room and kitchen look like the same paint-it looks pretty fresh[they probably did it to spiffy up and sell the property]