You've raised several questions here, all important.
1. Knife skills. Yes, you should work on your knife skills, and this is not as simple as you might think. You need to develop a good grip, probably a classic pinch grip, and a habit of clawing your left hand to hold the food steady with your fingers out of the way and the knife cutting such that it's guided by your left hand knuckles. (If you're a lefty, reverse that.) Easiest to learn, probably, is celery, but you must learn to chop an onion quick and clean. I strongly recommend that you watch Jacques Pepin: whatever techniques he demonstrates, whatever knives he might happen to have in hand, he always does everything cleanly in the classical French style.
2. Processors. A processor should not be used as a replacement for a knife. Until you are comfortable with a knife, certain things in large quantities will be cleaner and faster with the processor, but eventually you will not take it out for this purpose. But making a puree is unquestionably a raving pain in the tail if you don't have automation.
Somebody mentioned hummus, which is a wonderful example. To make hummus in a blender, you'll have to add quite a bit of liquid and you may end up with wet glop. To make it by hand is a labor of love requiring long pounding with a mortar and pestle such as you don't probably have. To make it in a processor, dump a can or two of high-quality chickpeas in the bowl, add a good squeeze of lemon juice and a good 1/4 cup or more of tahini, possibly a little minced garlic. Pulse and then run the processor until it's smooth. Taste, adjust a little bit as necessary, and it's done. Even with very ordinary canned chickpeas you will get better results than most hummus you can buy, and you won't believe how much cheaper it is. Now wipe out the bowl and do the same thing with roasted eggplant and you've got baba ghanoush.
Or try this: put four hardboiled eggs and a clove of garlic in and pulse a few times to break them up fine. Add some white wine vinegar, Dijon-style mustard, good pinch of salt and pepper, and a little parsley. You can also add a few anchovies, a bunch of capers, a little hot or mild raw pepper (the vegetable, I mean, not the spice), some shallots or onions, etc. Pulse a few times, then run it full-blast. As it runs, pour in some neutral oil in a thin stream (canola, peanut, corn, etc. oil). As soon as it starts to get thick, stop the motor, scrape it out into a bowl, and then whisk steadily and not too fast while you stream in good extra virgin olive oil. As you keep adding it, the sauce will get thicker and thicker, and eventually will be the consistency of mayonnaise. This sauce, basically sauce Gribiche, is an admirable (and salmonella-free!) substitute for mayonnaise in a great many applications, and believe you me, it is a raving horrible pain to make without a processor.
One more: Next time you cut up a chicken (a skill you should definitely learn --- saves money and produces great side benefits, like soup and the following), pull out the fat pads at the bottom opening of the chicken --- these are the big yellow wodges of fat around the sides. There might be one up by the neck, so pull that out too. Notice that the liver, which came probably in a little paper packet inside the chicken, weighs about the same as the fat pads. Put the fat pads in the processor and pulse briefly to chop up fine, then scrape them out into a small skillet. Cook over medium-low heat, stirring occasionally, until the fat is basically melted and not hissing (meaning the water has cooked out). Put in the liver, cut into lobes if it's all one big chunk, along with about 1/4 minced onion or better one minced shallot, as well as a generous pinch of salt and a little black pepper. Cook, turning the liver now and again, until the liver is just firm and just barely pink in the center. Remove the pan from heat, wait a minute or so, and scrape everything into the processor. Add 1 Tb butter if you wish (I always do). Pulse until basically chopped up fine, then run until good and smooth, scraping down the sides now and again. Scrape out the mixture and push it through a wire strainer, the finer the better. Pack into a ramekin, cover by pressing plastic wrap right against the surface, and put in the fridge. After 1-2 hours, remove from the fridge, wait about 10 minutes to let the chill off, and then spread on good crusty bread. Cost? Free --- you bought the chicken. Work? Minimal, because of the processor. Result? Excellent chicken liver terrine.
I could keep going.
The point is this: you are just learning to cook. You will want to play with recipes, try new things. You will find that there are lots and lots of recipes that involve a processor to transform a hideous chore into a brief zapperooney, as it were. The things just aren't that expensive these days, if you don't buy a really big one, and once you get used to it you'll find it remarkably easy and fun.
Incidentally, I'm assuming you've got a dishwasher. If not, skip the processor: handwashing the parts of a processor is horrible and potentially dangerous. If you've got a dishwasher, just rinse and whatever under hot water to get it basically clean, then put the pieces in the dishwasher and it's all done.
Automation, used intelligently, is a good thing.