There are lots of lenses that are better than kit lenses such as the typical AF-S DX Nikkor 18-55mm f/3.5-5.6G VR, for example.
That's not to say that the current 18-55mm "kit" lenses from Nikon and others are bad. Everything on the market from the major camera manufacturers is pretty good these days. It's just that some lenses, to borrow a phrase from Cajun cook and humorist Justin Wilson, are "more better" than others.
We could list a wide variety of different lenses at a wide variety of price points, but we still wouldn't know what lens(es) are better for the specific kind of shots you wish to create, much less which ones fall within or outside of your budget.
- Some are better for outdoor family portraits. Different ones may be better than others, depending on the number of family members involved, the time of day and weather conditions, etc.
- Some are better for senior pictures. Different ones may be better than others for different types of senior pictures. Indoors in a studio with fully controlled lighting? Outdoors in a variety of settings with a variety of natural lighting conditions? Head shots? Full body? Environmental?
- Some are better for newborn sessions. Different ones may be better than others for different types of newborn sessions. Cramped surroundings or plenty of room? WHat kind of lighting? Elaborate props or basic set with the newborn filling most of the frame?
In general, prime lenses perform better optically than zoom lenses do. This is particularly the case when a prime lens is at roughly the same price point as the zoom lens to which it is being compared. Normal¹ prime lenses costing only a few hundred dollars can often perform as well optically as zoom lenses costing thousands.
What one gives up with prime lenses is the flexibility of being able to alter focal length without changing lenses. This can be important when there are issues such as space constraints, or when one is shooting subjects that tire easily and may not tolerate the time required to swap lenses frequently during the shoot. It's also important with prime lenses to be sure and select an appropriate focal length for the task at hand. There's no "wiggle room" to get the framing right from a specific distance for a specific perspective like there is with a zoom! Even with the same focal length, different lens designs can make the resulting photos look very different. A 90-105mm Macro lens optimized for close focusing and flat field performance from edge to edge of the frame won't be as suitable for portraits as another 90-105mm lens optimized for longer focus distances and smooth out of focus areas.
Among zoom lenses, those with a lower ratio between their widest and longest focal lengths generally tend to be better optically than those with a wider range of focal lengths when both are in the same price range. A 17-50mm or 17-55mm zoom lens doesn't have to make as many design compromises as an 18-200mm lens does.
This answer to a slightly different question covers the differences between different types of lenses and why a lens specifically designed for a particular task can be better for that task at the expense of being less suitable for other tasks.
There are also many other things that can improve one's results with the same lens and camera.
- A good tripod and sturdy head with a remote cable, infrared, or radio release is one such thing, for instance.
- Good lighting and modifiers are another. Putting the right light on a subject reduces the demands placed on the camera and lens to pull something out of what is sometimes very little to work with.
- Good post-processing skills that draw the best results out of the combination of any particular gear and lighting.
Part of what it means to be a capable photographer is the ability to recognize what one needs and what one does not need in a particular piece of equipment to get a particular kind of shot. Part of that comes with experience and playing around with different bits and pieces to see what works. But a good portion of that can also come from studying others who have taken similar types of photos to what one wishes to create and learning how they got the results they did: not only what gear they used but even more so what techniques they used, how they lit the scene, etc.
The key for most of us to getting cleaner, crisper pictures is usually technique.
Determining exactly what is making one's photos blurrier than one would like is covered in this question: How do I diagnose the source of focus problem in a camera? The accepted answer has a lot of links to other questions here that cover many of the various reasons why images may not be as sharp as we'd like them to be. Most of the links deal with issues involving shooting practices and technique. Once issues with technique are eliminated, only then can we begin to consider that we may be pushing up against the limits of a particular lens or camera or other piece of gear.
When one reaches that point, it is really up to the individual photographer to select what is most appropriate for the kinds of photos one wishes to produce.
¹ A normal lens is one that is roughly the same focal length as the the diagonal of the camera's format size. For instance, the diagonal of a 36x24 mm full frame/35mm film camera is about 43.5 mm. Lenses from about 40mm to 55mm are considered normal for the 135 format.