Telephoto lens for a cropped sensor

Having a "hole in the middle" is more trouble than you could possibly anticipate as a newbie. Believe me, I did it & my hole in the middle was only between 55mm & 70mm.
For some bizarre "Murphy's Law" reason, everything you ever want to do seems to fall into that hole.

There is, of course, the inevitable GAS [Gear Acquisition Syndrome] that means we always really, really need just one more lens… but at least start with a one-size-fits-all lens that doesn't leave such a sizeable hole.

I eventually [after first buying 2 others that weren't as good] went for what I now call my guilty pleasure lens, the super-zoom Nikon 18-300mm.
It's not the sharpest knife in the drawer, but it does mean if I'm on walkabout I only need the one lens with me, rather than my whole bag. Focus speed, low light usability & VR are all better than you'd expect. Image sharpness is really OK & only softens right out at 300mm.

Of course, it's not as fast as the 2.8 zooms, but you end up needing all three of those, at about 7 grand if you get the Nikons.

It would be beneficial to consider a lens in the normal range (~28-40mm for APS-C). Once you're familiar with normal, it should be easier for you to figure out what to do for wider and narrower.

Options to cover normal include:

• 35mm prime.
• 18-55/3.5-5.6 "kit" lens. It has since been revealed that you already own this lens.
• 24-70/2.8.
• 24-120/4. This is the usual travel/walkabout option.
• Tamron 35-150/2.8-4.
• Etc. There are simply too many to list them all.

For travel and landscape, weight is often a concern. Many simply go for a travel/walkabout lens (noted above), but that doesn't work if you need greater "reach". Some might go for a superzoom, as Tetsujin describes.

You pretty much have to weigh the pros and cons to decide for yourself what will fit your needs. Reasonable telephoto options include:

• Prime lenses
• 55-200/4-5.6
• 70-200/2.8
• 70-300/4-5.6
• Others.

Many stores will let you try lenses to see how they handle (may be problematic with COVID-19). You can also rent (or borrow from friends) to try them in more realistic settings.

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Note: Crop factor is used to compare different sensor formats. Since you have only one camera, there is no need to multiply focal lengths by crop factor. Doing so tends to be more confusing than helpful, especially if you are not careful about specifying whether a stated focal length is the actual focal length or the full-frame "equivalent" focal-length.

For instance, you state that a "Sigma 50-100mm 1.8... would be 75-150mm on my camera." That is technically incorrect. It would still be a 50-100mm lens on your camera, but would produce images that look like they were taken with a 75-150mm lens on a full-frame camera.

There is also ambiguity about whether any isolated focal length is an actual or adjusted value. For instance, you ask, whether "100mm would be good". Are you referring to an actual focal length or an adjusted full-frame "equivalent" focal length?