I'm playing 3 Houses foe the first time, finished Verdant Wind and just started Azure Moon. Most of these builds are things I've heard other people talk about or things I have thought of but haven't had the chance to actually test. Btw I'm playing on Maddening, these may be overkill on Hard.
Dimitri
One really common build you'll see in Fire Emblem is Vantage + Wrath with the goal of the build being to kill enemies during their turn before they have the opportunity to even hurt you but it requires you to be at half health or less, so if you fail to kill you are putting your characters at risk of death. But Dimitri gets Battalion Wrath at C authority and Battalion Vantage at A authority which allows Dimitri to use this combo without him actually being at low health. I think what people do purposely not repair low health battalions and keep them around for Dimitri to use.
Another build I see people do is dodgetank Dimitri. Sword Prowess 5 gives 20 avoid and Dimitri's sword strength helps him get A+ swords more quickly. If he qualifies for the dancer he can get the skill Sword Avoid +20 (although this means giving up your dancers for a more effective dodgetank). Post timeskip his personal skill will give him +20 avoid while at full health simiar to Ferdinand. You can train him in flying to get Alert Stance which gives +15/+30 if Dimitri uses the wait command. But I heard there's a period of time where he can't be instructed so I imagine you probably have to make him a Wyvern Rider so he has the opportunity to raise his flying rank. You can also give him a battalion with a lot of avoid and the avoid ring to give him even more avoid.
Since the first build takes 2 skill slots and the last one takes 2-3 skill slots I imagine you can combine them to make a Dimitri that's really good at killing things during enemy phase.
Dedue
I don't know what people build on Dedue but I intend to make him an Archer for hit +20 to help his accuracy issues, then send him through Armor Knight and end in Fortress Knight with a silver shield to double down on his physical defense and hopefully he'll have enough defense to barely take physically damage.
Ashe
I plan to make Ashe go Fighter -> Brigand -> Archer -> Sniper and he'll probably spend most of his time poking enemies from afar or spamming Hunter's Volley. I first focus on Axes in the early game so he can qualify for Brigand, master it for Deathblow and then make him a Archer and then Sniper which will be his main classes. I intend to have him end with Bow Prowess 5, Bowfaire, Deathblow, Hit +20 and either Strength +2 or Bow Crit + 10.
Mercedes
I plan to just have her focus on faith and reason, send her through the mage classes and have her end in Gremory, nothing special really. I feel like she doesn't care for any of the physical class mastery abilities except maybe Darting Blow but with her awful Strength and lance weakness I'm not sure if it's actually worth training her in lances and flying to be a Pegasus Knight for long enough to master the class unless you have her be somebody's adjutant for the entire time. You'll definitely want her to to master Mage for Fiendish Blow.
Annette
Kind of the same thing as Mercedes except she's neutral towards faith so getting her to qualify for Gremory will be a bit harder. I first taught her in faith so she could learn heal and then prioritized authority on her until she reached C+ and learned Rally Speed which stacks with her personal skill which is basically Rally Strength so she can give +4 speed and strength to one ally.
Felix
I plan to go with a brawling build and stack enough damage for Felix to hopefully be able to kill foes on player phase before the have the chance to counter attack. His class path will be Fighter -> Brigand -> Brawler -> Grappler. Brawler is optional, I'm only having Felix ]turn into one because it earns slightly more gauntlet experience than Brigand. Once I master Grappler on him he will be spamming Fierce Iron Fists a lot. His abilities will be: Gauntlet Prowess 5, Fistfare, Deathblow, Fist Crit +10 and Strength +2.
Ingrid
I'm first having Ingrid focus on Axes and mastery Brigand for Deathblow. Then I'm making her a Pegasus Knight, then a Wyvern Rider and lastly a Falcon Knight. Most people prefer Wyvern Ingrid as opposed to Falcon Knight Ingrid but I wanna do it cause it's supposed to be her canon class. Though it does make more sense to go Wyvern Lord since if you're getting Deathblow you're already going to be levelling up her Axe rank anyways. On top of that, Wyvern Lord does improve her her meh HP Strength and Defense.
Sylvain
I'm first focusing on axes to make him a Brigand for Deathblow, then having him focus on lances to try and get Swift Strikes ASAP and putting him into Cavalier then ending in Paladin. I've heard Dark Knight Sylvain is a thing. He has a strength in lances and riding and a budding talent in reason that gives Black Magic Avoid +20. His magic stat isn't good but it isn't terrible either and he gets an axe magic combat art at A axes.