It can be a bit daunting to wrap your head around specialists and citizen efficiency. In general, I will use the automatic settings, mostly default or production, and it works well enough for me. Sometimes you will want to manually select tiles to be continuously worked, for example, say you settle the natural wonder Mt. Sinai (which I believe provides 8 faith per turn) with your 2nd city. The game will probably want to work growth tiles, but you (wanting to be the first to found a religion) can manually select Mt. Sinai to be worked and ensure you get a religion quite quickly. Or maybe the city will want to work a farm beside a river and grow in 6 turns but you want to work the farm on a hill by a river to get extra production but the city will not grow until 9 turns.

I always select manual specialist control in my cities. From what I've experienced and read about, on the higher difficulties you will mostly want to focus on science specialists. Great scientists, engineers, and great merchants all come from the same "pool". This means that for every GS, GE, and GM you spawn, the next one will cost more. The GM is the weakest of the three so you shouldnt really work them because GS and GE are far more useful in nearly all circumstances. Great writers, artists, and musicians come from separate "pools" and each one you produce only increases the cost of the corresponding great person.

Often you will want to keep great people for specific things. I usually build academies with GS until around turn 250, then keep the rest until 8 turns after I build research labs in my major cities.

I usually keep any GE I have to rush build a wonder in a city unless I have a very low production city that I think could benefit from a manufactury.

With GS and GW their bonus comes from the previous 8 turns of science or culture. That is why you generally save them if you are going to complete any science or culture buildings. If you manage to come first in the worlds fair project, you receive a huge culture boost so when you receive the bonus from a GW it is far larger than usual.

There is so much to say and im on my phone!. A good place to check is the civfanatics website. It can be a bit disorienting searching through it but if you have ever used the reddit search fiction you should be fine.