Lens factors that give deeper depth of field

The longer a lens is, and the larger its aperture setting (limited by its maximum aperture), the less depth of field and the more background blurring it will give. (See: How can I take a photo with everything in focus with my DSLR?)

Portrait lens

Lenswise, I'd actually say that what you're asking for is the opposite of most peoples' definition of a 'portrait lens'. Most of us consider a lens that can yield a thin depth of field with a large aperture (f/2.8 or larger [smaller f-number])--either a short telephoto prime (e.g. an 85mm f/1.8), or a telephoto f/2.8 zoom to be a portrait lens. And these lenses are not what one would typically use for landscape shooting, which is a wide angle (short focal length) lens.

Choosing a first lens

I'd suggest first using the kit lens that comes with the camera body before deciding whether you need a fast prime lens. It can stop down to f/8 to give a deep depth of field just like a fast prime would, covers a number of different focal lengths, so you can get a chance to see what focal lengths/working distances you favor, and can be used with a flash, just like any other lens. It can also go wide enough to be used for landscape shooting. And--more importantly--kitted with a body, it's the cheapest lens you can find, so you aren't spending much on your "training wheels" lens.

Buying lenses as a newb is a chicken-or-the-egg problem. To know what lens you want, you need experience with lenses, but to have experience with lenses, you have to get a lens. The kit is a nice, low-cost solution to getting some experience, even if it won't be your final lens. And all lens decisions are highly personal--everybody has different subject matter, different working distance preferences, different compositional needs, and different budgets so there are no object "best" choices for any given thing. (See this lens upgrade answer).

Interchangeable lenses tend to be lousy general-purpose all-in-one tools, but great specialized task tools. Chances are you'll eventually want two different lenses: one for landscapes, and one for portraits. This is why interchangeable lens cameras are so much more expensive--just getting the camera and a lens is only the beginning of building your camera system.

Other DoF factors

But also realize that background blur isn't determined solely by the lens. There are a number of factors that affect how much out of focus blur you can get in a scene. Your working distance, and the distance from the subject to the background may actually be even more important.

Bigger Sensor → More Blur

The size of the sensor also affects the amount of background blur you get. So, if you're very concerned about DoF, you need to consider that as well, not just which camera will fit the lens you want. For example, micro four-thirds cameras have sensors with a 2x crop factor (i.e., how much smaller the sensor is than a frame of 35mm film), so they're better for attaining a deeper depth of field than an APS-C sensored camera, although not as good as your Coolpix is.

You're currently shooting with a 1/2.3"-format sensor in your Coolpix P510. This is very small. It has a 5.6x crop factor. dSLR and mirrorless interchangeable lens cameras often have 1.5x APS-C sensors in them--making them roughly 3-4 times larger than the sensor you're working with. As a result, the lenses will be 3-4 times longer. To frame exactly the same way as you would with your camera, you may be using a longer lens, or working closer to your subject. Both of which reduce your depth of field and increase the amount of background blur. So moving to a larger-sensored camera inherently means more background blurring. You won't just see it in macro shots any more.

Consider a flash

Your main tool for increasing the depth of field with a larger format camera is going to be using a smaller aperture and a shorter lens. But too short (say, below 50mm), and you can distort someone's face. And too small an aperture, and you may not have enough light to prevent motion blur because you could force the camera to use a slower shutter speed. So, you may also need to add flash to the scene to get the light levels bright enough. This is why off-camera lighting is the go-to technique for professional portrait photographers. So, you'll also want to make sure whatever camera you choose has a flash hotshoe.

See also: What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera?