What are the best lenses for outdoor portrait photography and for landscape photography?

For either Nikon or Canon? I'm thinking of purchasing a Nikon but may do Canon.

Most professional portrait photographers will tell you to use a moderate telephoto when doing head shots. While not engraved in stone, a 105mm on a full frame (FX) is the preferred focal lenght. This is about 2 1/2 X time the corner-to-corner measure (diagonal measure) of the film size or digital sensor size. If the camera is of the APS-C with it's smaller sensor (crop factor 1.5 or 1.6) then use a lens centered on 65 thru 70mm. For landscape, usually a wide-angle lens is preferred. I would use a 28mm if the camera is an FX or a 18mm if a compact digital (DX). We're talking art and their are no rules in art, you are free to follow your heart.

In order to give recommendations, you need to state which format and style of camera. Both Nikon and Canon make APS-C and full frame DSLRs, and now they both make mirrorless cameras, which use different lens mounts.

Generally speaking, the best landscape lenses by these two brands will depend upon your personal style. For most, the Canon 11-24 and the Nikon 12-24 are the best for landscapes. However, if you're using a crop sensor camera, you won't be able to take advantage of their ultra-wide angle of view, which is why landscape shooter will always use full-frame cameras. Not that there aren't wide-angle lenses for APS-C cameras, it's just that none are of a high quality like the two aforementioned lenses. The only options for the APS-C format are the 10-? Mm zooms such as the 10-22 or 10-20.

Portrait lenses are any focal length between 85mm~200mm. Depending upon your needs and your personal style, you'll either feel that a prime like an 85mm f/1.4 Otus from Zeiss, or a Canon 135mm f/2 are among the best primes for portraits. I think that most would agree that the 70-200 f/2.8 is probably the best zoom, and definitely the best lens to have among the other options. Most photographers will buy the 70-200 f/2.8 first, and then depending upon they personal style (or needs of their clients) will add an 85mm or a 135mm to their kit. If they're shooting APS-C, then the 50mm f/1.2 or 50mm f/1.4 will be one of the most preferred primes to add to their existing kit.

These lenses are the best because of a few reasons. First, they provide the correct angle of view. Second, they're all very well made and sharp. Thirdly, regarding portraits, these lenses provide a very large aperture that allows one to blur out backgrounds. Primes are desirable specifically for this very reason. They'll have even larger apertures than a zoom allowing one to create interesting effects by utilizing a super-shallow depth of field.

You don't mention a budget. But since you asked for the best, I've given these suggestions.

The 18-55mm kit lens that comes with most entry-level dSLRs will do for both portraits and some landscapes. For more background blurring and pulling-in power, you may want to get a longer short telephoto lens with about 70 or 80mm and a maximum f/number lower than f/3.5 for shooting close-up portraits. For more expansive landscapes, you may want to go for a wide angle lens that's lower than 18mm. To learn more about how to look at lenses so you'll know what to buy next, see this.