Of the two, learning better knife skills is going to help you the most. You also want a good sharp knife. It also takes a little practice. I've gone from taking forever to cut up an onion to having one totally cut up in about half a minute.
There's also some good food prep books and videos like Wusthof's and Henckel's that are pretty good. A food processor has limited use. It comes in handy at times. But there's no replacement for good knife skills.
For example, you could use a food processor to dice up bell peppers. But you still have to cut off the top, cut out the seed portion and cut the stem out before you stick it in the food processor. And by that time you already have most of the work done. And if you throw it in the food processor, now you've dirtied up another dish that you'll have to clean later, adding more time.
Don't go out and buy a whole cheap knife set. Instead start off with about 2 good quality knives with a classic French style handle. A good 8" chef/cook's knife and a good pairing knife will take care of most of your needs. It is far better and less expensive in the long run to get good quality knives instead of getting a bunch of knives or a full set.
Wusthoff and Henckels are both pretty good. One mistake many people make is they go for the "ergonomic" handles. Many brands when they make the "ergonomic" handles don't properly design the handles for a proper pinch grip. Instead, the knife gets designed to feel well in the hand when a person uses an improper grip where they hold the knife somewhat like a person would hold a sword. Those types of "ergonomic" handles will feel great with an improper grip. However an improper grip with lots of chopping will likely have your wrists hurting. And when you switch to a proper pinch grip, the ergonomic handles often feel odd in your hand. A classic style handle will feel good when gripped both ways. It may not feel as wonderful as the ergonomic handle when you use the improper grip, but it will still feel comfortable for both methods.
The thing I see most people struggle with when cutting is they try chopping through everything instead of slicing through it. Now I'm not saying chopping is bad. For some things you want to do a quick chop.
A knife typically cuts easiest when it slices through food. By that, I mean there is both vertical AND horizontal movement of the knife. A lot of people try just pushing the knife down vertically through the food. Think of the knife like a papercut. You can't cut your finger easily with paper if you just press the paper up against your finger. However, if you slide the paper then the paper will cut you. It is the same with a knife, that sliding movement helps the knife cut better.
Learn to take the rounded tip portion of the knife and rest that against the cutting board. And bring the back portion of the knife up to form about a 30 degree angle to the board. Then slide the tip of the knife away from you as you bring the back portion of the knife down to the cutting board to cut. After finishing the cut, just reverse the movement. Bring the tip of the knife back towards you as you raise the back of the knife up to make the next cut.
A good way to visualize it might be to think of the back of the knife attached to a pin on a rotating wheel. So the back of the knife is making this circular motion while the front of the knife rests on like a sliding track. Or imagine the wheels of an old steam locomotive and that bar going from the wheels to that piston shaft. That's the motion you make with your hands.
There's about 4 major things you want to learn how to cut.
1. Learn how to properly cut an onion (this can be applied to shallots and even garlic with the skin removed). You can smash then mince garlic. But if you have a steady hand, cutting garlic like an onion is better. Because when you smash then mince, the garlic wants to stick to the knife blace. And you keep having to push the garlic off the knife blade to mince it. The onion method is faster.
2. Learn how to properly cut a pepper (the method for a bell pepper is somewhat similar to hot peppers)
3. Learn how to cut a tomato. You can cut the top and bottom off. Then cut through the side and start unfolding the middle as you cut away the center, removing the tough center and all the juice and seeds for just the meaty portion of the tomato. If you want, you can then cut the skin off similar to filleting a fish.
4. Learn how to properly mince herbs, etc
5. Learn how to slice and chop stuff like carrots, celery, etc.
My favorite knife design is a Japanese style of knife. I've seen the design sold by at least 3 different Japanese companies. It cuts the easiest of any knife I've used. That design gets sold under three different labels: Hattori HD, Maruyoshi HD and RyuSen Damascus. It is basically the same knife but sold by 3 different companies. Hattori and Maruyoshi call it their HD line. And RyuSen calls it their Damascus line.
When you learn how to properly cut an onion and make that horizontal cut. That's where you can really tell the difference between the really good sharp knives and the dull or so-so designed knives. When you make that slice and you have to push or saw instead of making one easy slicing motion, that knife is too dull or has too thick of a blade.