Suppose one wanted to become an expert in a particular genus of insects, with many species spread over a large geographic area. Examining specimens and learning the diagnostic characteristics is necessary. But museum specimens are not all concentrated in one place; they are held by many different institutions in many different places.
So, first one would have to find out which museums hold specimens of the genus in question, and secondly would have to gain access to said specimens. How is this normally done?
Often museums have a geographic focus of local specimens from their surrounding area e.g. if you want to learn about stuff from Hawai’i, the Bishop Museum is your best bet. Some museums have also had historical focuses on specific regions from further across the world (my examples come from botany, but I believe the concept is the same in entomology) with Kew having done a lot of collecting in Central Africa, Missouri botanical garden doing a lot in Madagascar and Panama, The Field Museum in Guatemala etc.
Your best bet is to look at monographic work which has previously been done on your taxa of interest and see where the deposited their specimens or where they traveled to the view specimens. This will usually be in their methods. Also just ask somebody who is broadly knowledgeable about the group or even a similar group and they can probably give you some leads.
Ultimately, you’ll probably have to do a fair amount of traveling if you want to see everything in person.
Also, try going on GBIF and see if you can search for your taxa and region and see if you can find a breakdown by institution. Use the record type “preserved specimen” The limitation here is that not everything may be on GBIF so you could be missing some major collections.
Many natural history museums now have their catalogue information online. For example: https://collections.nmnh.si.edu/search/ento/
If you find that particular museum has specimens you need to examine, the usual method is to write to a curator in that department, briefly explain the need, and ask what their procedure is for gaining access.
Asking the curators of the museums is the best starting point. First to learn whether they have some specimens that you’re interested in and secondly to figure out how to access them. If they have an online catalogue then this of course makes it easier for you to figure out if they have interesting material. If you’re not associated with another established institution or museum and if you can’t prove that you are seriously working on this taxon (by publications for example) then it might be a bit harder to access specimens because museums not always show their collections to everybody. Some museums might be more generous than others (collected specimens that are properly ID’ed are very valuable after all). From what I’ve heard, museums also usually only send specimens to other museums or institutions to make sure that the specimens are treated well. Hence, if you’re not associated with a museum or institution don’t expect them to send you specimens.
It should be possible to learn features from a description. I’ve often gone back to the original descriptions to learn the defining characteristics. Every species should have a type description published some place. I’ll often use Google Scholar, but note that some older species descriptions have had changes in Genus, so use the species name to search. Keep in mind that even with a physical specimen a microscope may be essential, and some features can only be seen by dissection.
If you’re a scientists who needs type specimens for the work you contact workers of institute they’re stored in and them there’re formal stuff to get it, but, not everything can be achieved that way and for some specimens you have to visit the location and work with them on spot. If you just need those specimens to learn something you need to contact the curator and visit the collection.
I second that starting with an internet search of GBIF or other taxon-specific portals is a good way to figure out which institutions have major collections. This will work for most taxa and many institutions have their collections online to some degree now.
There may be specific taxa or places that aren’t as digitally accessible though, so YMMV.
Once you’ve got the institution you’re interested in, contacting the section curator and expressing your interest and reasons will be the way to go.