The expected use of the gun plays a huge part in the design of the sights. In general, the longer the expected effective range of a gun, the more precise and complicated a gun's sights are likely to be. Before rifling was commonplace, sights on even infantry muskets were very rudimentary. Rear sights were often omitted from the gun, and when they were present you'd often see them in the form of a simple fixed V-notch. Front sights were also fixed as well. While this would seem like a poor idea - the gun can't be zeroed once it's manufactured and precise aiming at a distance is going to be difficult - the reality of it was that, in an era of smoothbore muskets, effective firing ranges were short enough that even a simple front sight (similar to modern sporting shotguns) would be more than enough for a soldier in formation to fire effectively. However, with guns intended for more precise shooting, you start to see more effective sights on the gun. The Baker Rifle, for example, was a contemporary of the Brown Bess musket designed for more precise and longer-distance shooting with skirmishers. Because of that, we see significantly more effective sights on the Baker than we do the Brown Bess - not only do all Baker rifles have a complete front and rear sight (as opposed to the Brown Bess' lone front sight/bayonet lug), but the Baker even had a flip-up rear sight for longer distance shooting. By the time of the Civil War, where rifled muskets firing the Minie ball are standard issue, effective ranges have extended out to several hundred yards, and sights on rifles generally became more advanced to make use of this. The Civil-War era 1861 Springfield had sights with flip-up settings for 300 and 500 yards, and the British Pattern 1853 Enfield took this even further with sight settings in 100-yard increments up to 1,000 yards.
With handguns, this is even more apparent. Early handguns often had no sights at all for much the same reason that sights were simple on general-issue muskets - the effective range of a smoothbore handgun was so short that aiming down the length of the barrel was often enough to ensure a hit at any distance where it'd be effective. Even as rifling comes along, a lot of factors would work against pistols to keep effective ranges short. The low velocities produced by black powder weapons out of a short barrel ensure that the effective range of a round is going to be short, even if it's more accurate than a smoothbore. For the user, a very short sight radius also makes aiming at distant targets significantly more difficult, and without a stock, it's significantly harder to hold the gun steady enough for consistent long-range shots. Because of that, there's not much reason to install sights of the same quality as the infantry long arms, as the gun's never going to shoot that far.
That being said, we do generally see sights improving over time on handguns. Front sights became standard fairly quickly, but with short effective ranges, we end up in a situation much like the earlier muskets - aiming off of the frame and a front sight post was considered "good enough." Although sights on handguns would improve over time, we generally see even to this day comparatively simple sights on handguns. Rear sights became standard and sights were improved, but with the short effective range of even modern handguns for many of the same reasons stated above, very simple sights that aren't always adjustable are often considered good enough for the intended role.
The one anomaly to this is some of the early "modern" smokeless powder semiautomatic handguns like the Mauser C-96. Being developed at a time where these guns were some of the first to their field, the specifics of how they were going to be used hadn't been completely fleshed out yet, so we see a lot of cases of handguns with adjustable sights (in the case of the C-96, 500 to 1,000 meters depending on the model) often paired with some form of a detachable stock to turn the gun into a compact carbine. However, this fairly rapidly fell out of favor. Although compact carbines would survive (the Cold War-era vz.61 is a good rough analog for the C-96), they reverted back to simple sights, as users aren't generally firing such weapons at any distances where more precise sights would be necessary.