Landscape photography typically calls for the entire scene to be in reasonable focus. Also, if you have to look around to take in the scene, then it probably calls for a wide angle lens. Wide angle lenses also help stretch the sense of depth of an image. Focus speed isn't an issue... it doesn't matter how snappy the focus motors may be. It turns out most kit lenses have all the attributes you need to get started in landscape photography.
You could spend more money and buy a new lens, but you probably won't notice any benefit from that until you gain more experience.
There are other things you may want, such as:
1. A tripod - this will let you shoot at very low ISO settings in any light (especially considering some of the best light is during the earliest or latest hours near sunrise or sunset when light is lower.) This gives you a stable platform and camera movement won't cause any blur.
2. A circular polarizer - but careful with this. This is a filter which threads onto the front of the lens. It's selective about the polarity of light that can pass through the filter depending on how you adjust it. It turns out there are a LOT of reflections in a typical image -- especially off foliage. By polarizing the light, the green foliage looks greener. The sky seems to be a richer shade of blue. White clouds stand out more. It has an overall effect of helping to intensify the color of an image. But... if used at very wide angles, you get a darkish banded region in the image where the polarization effect is strong... and that's surrounded by areas where the polarization is weaker. Another side-effect is that it does have the effect of reducing the total amount of light and that increases the exposure time... which is another reason why it's good to have that tripod I mentioned in #1.
3. A "neutral density" filter. Think of these as sunglasses for your lens. All they do is cut the light. It turns out they come in strengths... if you were to only buy one, then a 3 stop neutral density would be a good starting point. By cutting the light (beyond what you could do by just reducing the aperture opening in the lens) you can further extend the amount of time that the shutter can be open. This is particularly desirable when you have shots that include water. Waterfalls get smoothed out and it helps convey the sense of motion -- that the water is flowing. A 10 stop neutral density is even more extreme... but slightly trickier. You can't see well enough to focus through a 10 stop ND filter, so you have to focus with the filter removed... then switch off the auto-focus, attach the filter, then take the shot. Again... these extended shutter exposure times means the camera cannot move during the shot... so you'll need that tripod I mentioned up in #1.
4. A "gradient neutral density" filter. There's a variation of the stand "neutral density" in which half the filter is tinted... half is clear. The filter may transition from clear to tinted very rapidly (which they call a "hard" edge) or gently (which they call a "soft" edge). These allow you to darken the sky... but not darken the foreground. The result is a more balanced image. The trick to this filter is that it typically is a rectangular filter and it slides into a filter holder ... this allows you to decide exactly where you want the tint transition to appear in your image.
All of these will be less than the cost of a new lens.
Don't go run out and buy them all... I suggest you just buy a tripod and perhaps the circular polarizer. If you find yourself taking lots of images that include flowing water... then a neutral density filter would be a great addition. The "gradient neutral density" filters are a bit more advanced... hold off on those until you're comfortable with the camera and tripod.
Portraits, on the other hand, are completely different. A longer focal length lens with a lower focal ratio can allow you to create a tack-sharp subject with a beautifully blurred background. It's a very attractive look. Ideally this would be an 85mm or longer lens. Shorter focal length lenses (e.g. 50mm) get a weaker version of the effect.