I am not one of the masters who have posted here, but since I work Caravans a bit differently I'll say what I do.
Not the Great Library
1. The trade bonus is nice, so I always deliver into Demand, at max trade arrows. The ongoing route arrows seem more important - they are like roads. I rehome Caravans freely to get the Demand to work between the cities I've selected.
2. I examine all cities each turn, and IPRB Caravans at 10,20,30 and 40 nearly always. If you are new to this, try my "Rule of 28". When a city is building 7 production, and it has completed 28, change to Diplomat or Archer (30), Rush Buy $4, then switch back to Caravan. That is what we mean by IPRB Incremental Partial Rush Buy: it saves an whole turn and is a very efficient use of $gold.
3. I play conquest, Emperor, and I build an SSC (Super Science City) with HG, Copernicus and Shakespeare etc, usually my Capital. All Trade routes go to or from that city, until later when I may establish a route directly between two other cities.
4. Once in Monarchy, I'll try to get one trade route going between my SSC and the next city that I'm favoring to grow. This is stress, because I'm desperate to build Wonders, racing the AI for Hanging Gardens. No trade route until that is done. At King Level, the Pyramids is the first Wonder.
5. I choose a city to favor, rename it with a $ such as Nottingham$ so I can see it on all the report screens. I work to get 3 trade routes there, then move on to the next favored city.
6. Once in Republic, the trade arrows are doubled and I'm enough ahead on Wonders to put progressively more effort into Trade Routes. In the current Gotm27, I've built 12 wonders at a cost of 5*4 + 5*6 +2*8 = 66 Caravans, and have 14 Trade routes going, 1460AD, so most effort has gone into Wonders.
7. If there is an AI city that is convenient, or if I'm suffering delay getting Science, I'll deliver into demand far away there. A ship chain is helpful. If they cancel the demand, I bring the Caravan back and use it for the next Wonder.
8. Some of the Civ2 Gotms were Fundy. In that case, Trade Demand delivery bonus is THE way to get Science.