How do you change camera depth and close-up cutoff?

So one thing is if an object is too far away or too big, it will get cut off and disappear, and another problem is when you zoom in really close to an object, parts of the object disappear before you actually get to the surface, which is a problem mainly for small objects. How do I change these things?

This is all relative here in Blender; as soon as you start to mess around with thous and mm you’re asking for such troubles. Keep current model you work on Grid sized, keep to needed proportions not the absolute metric or imperial measurements. You can scale appropriate for for external applications after you have finished modelling.
One Blender unit equals to 1m is what is important for Scene lights and simulations like fluids, cloth where real world physical values need to be considered in relation to the time or luminance.

Clipping close and far values are set separately for 3D view ( N panel) and Camera, Data tab if Camera is selected.
Close - Start default value 0.1 is reasonable - if it has to be set smaller (due to object being set mm range) you’ll get certain view artefacting going on.
Far - End value can be increased without running into such artefacts. If it’s too high though you might start to see mesh being thorn apart. All this happens due to how floating point values translate into binary ones and zeroes. You want to be in a ‘safe zone’, close to the coordinate starting point.

If you work on imported from other applications mesh (which can be huge in Blender), scale it down 0.001 or more, apply scale. Move it to the coordinate centre. Scale back before the export.
Hope helps.

when working on stuff up close, with the item selected press the number pad . (period) then number pad 5
when in orthographic view(non perspective) you can zoom in really close
if the zoom slows down hit the period again to reset zoom to selected

The so-called “clip range” is a parameter of the camera. Only objects within this inclusive Z-distance range of the camera will be rendered, or considered by any rendering calculations. If objects are disappearing when you get too close to them, they’re being clipped by the near-distance parameter. If they vanish when they get too far away, the far-distance. One of the options that you can set for a camera is one that will draw a line in front of the camera which shows you its clip-distance settings.