Special classes and route order: Start with a faster route to expand later optionsMinor spoilers
Play Dietrich, Leda, Theodora, then Cai. Inherit map access and side quests before handling the more time-consuming exclusive systems.
Original video: “Fire Emblem: Fortune's Weave — Special Advanced class unlocks, recommended route order, and basic development tips”; Bilibili creator Dr__W. Watch the original
I recommend 
Dietrich → 
Leda → 
Theodora → 
Cai. Use the quicker routes to reveal maps and finish ordinary quests, leaving time for the later two routes' more turn-intensive exclusive activities.
Start with Dietrich to carry over map and side quest progress
I recommend Dietrich→Leda→Theodora→Cai. World map access and completed regular side quests carry over from the first route. Dietrich has a strong starting team and a simple, time-saving exclusive mechanic, making him suitable for learning the game. Opening the map and clearing side quests first leaves more room for routes whose systems consume more turns.
Unlock class options early
Dietrich's route unlocks 
Guardian, whose Strength, Defense, and Resistance growth and white magic suit front-line use. 
Blacksmith offers another early-to-mid-game choice for developing both Strength and Magic. These qualifications let later routes go beyond simply advancing from their starting classes.
Second, 
Leda's exclusive activities involve some detours but relatively little extra turn pressure, with a strong early team. Finish map areas and regular side quests missed before, while unlocking 
Dancer.
Leave turn-intensive exclusive systems for later
Theodora's and 
Cai's mechanics both consume extra turns. On a first run, unfamiliar timing makes it easy to miss other content while handling them. Inherited maps and side quests remove one burden. I play the queen before Cai so her 
Cataphract is available for him.
Cai can add extra mount growth, another reason to consider cavalry and flying classes on his route. I usually do not recommend them on the other three routes. 
Io and 
Alexandra bring special mounts of their own and can use them when recruited on other routes too.
Compare mount growth alongside class changes
If you do not mind similar builds across the party, physical units on Cai's route can consider 
Charioteer. Its two horses provide more mount growth, and the class skill raises its growth rates to advanced-class levels at level 35. There is no need to reclass solely for a higher tier. Magic users can use Black Pegasus or black Ornius mounts for Magic growth. Check actual growth rates before deciding on an advanced class; a higher class tier does not mean every growth rate is higher.