I would go with bladesinger. As far as levels go it depends on how much magic you want keeping in mind that the more levels in BM you take the slower your spell progression is.
It depends on what options you want, how you want to play it and what your abilities are. Here are a few options:
Options:
1. High Strength and High Wisdom: Heavy armored Temepest Cleric. Either 4 or 6 Battlemaster levels, the rest Cleric. If you go 4 levels pick up Hex, Booming Blade and Green Flame Blade from Magic initiate, alternatively start as a high Elf for Booming Blade. You can also look at gift of the chromatic dragon. This build has only 1 attack a round but it is very Gish and you want to boost booming blade damage to be as high as possible (Hex+Metalic Dragon+Divine Strike). You can maximize Thunder and Lightning damage with CD and move the enemy every time you lightning them. If you take 6 levels in Battlemaster you have multiattack and would use it instead of a cantrip.
2. High Strength and High Charisma: Battlemaster 6, the rest Paladin: Divine Smite, a bunch of smite spells. I like Vengence and Conquest for this. Wrathful smite and shadow touched with Cause Fear if you are conquest. I also prefer a Goblin-Conquest for this build to be able to disengage as a bonus action after you have frightened the enemy.
3. High Dex and High Intelligence: Battlemaster 4, Bladesinger X: Any race works well here. This is the combo that has the most flexibility you can be a very powerful melee character and a very powerful fighter. Again Goblin for disengage combined with Bladesinger Extra Attack and Booming Blade is nice. This iis the combo that has the most flexibility.
4. High Strength and medium Intelligence: Battlemaster X, War Wizard 4: Much more of a fighter than a Wizard but War Mage gives you the spells you want as well as the reaction +4 to saving throws (and +2 tto AC once you run out of shield spells). I like a Variant Human for this build picking up the heavy armor master at 1st level.
5. High Odd Strength and High Charisma: Battlemaster X, Shadow Sorcerer 4: Again more of a Figther than caster. With an odd strength would start as a sorcerer here for the charisma save with Strength of the Grave and then take heavy armor through a feat while boosting strength. Lots of Races work really good here. Damphir to be able to walk on ceilings (get a reach weapon and attack from the ceiling). Shadar Kai to use blessing of the Raven Queen and get resistance to damage. Half Orc which gives you relentless endurance and then the ability to do it again through strength of the grave.
6. High Dex, High Wisdom: Battlemaster 6, Ranger X. Any Ranger subclass works here. Fey Wanderer seems to be the most Gish build to me if you have a high Charisma. Dragonborm with the Dragon Fear feat works particularly well/
Six character builds that combine martial abilities and spellcasting, including a heavy armored Temepest Cleric, a Battlemaster/Paladin with high strength and charisma, a Battlemaster/Bladesinger with high dex and intelligence, a Battlemaster/War Wizard with high strength and medium intelligence, a Battlemaster/Shadow Sorcerer with high odd strength and charisma, and a Battlemaster/Ranger with high dex and wisdom.
Each build utilizes different subclasses and races to optimize their abilities, with varying focuses on spells and combat.
It depends on your abilities at 1st level and how many levels you want to devote to a "dip" With that in mind here are the best options in order IMO:
1. High Intelligence (15+) 2+ levels: Bladesinger is the best dip assuming you go at least 2 levels. This character can go anywhere from 2-20 and will outmatch any other option at any level. The only downside with a Bladesinger is you are actrually worse because of your fighter levels. A straight non-multiclassed bladesinger who selects spells specifically for melee is going to be a better Gish than a dex-based Bladesinger-Battlemaster. So it is basically when do you want to jump off.
2. 13-14 intelligence 2 levels: War Magic Wizard is awesome as a multiclass Gish. If you have a high intelligence Bladesinger is better. If you have a low intelligence you can't be a Wizard, but if you have a 13-14 a 2-level War Magic dip is awesome. +1 or +2 to initiative, +4 to saves, +2 to AC once you run out of shields.
3. 13-14 Intelligence 2 levels: Diviner gives you the spells you want and portent. This is not as good as War Magic though.
4. 13+ Wisdom: 1-2 levels: Cleric. Any Cleric will work for a dex character but Arcana and War Magic lean into Gish the best IMO.
5. 13+ Charisma, 4 levels: Sorcerer. Any Sorcerer would work, Shadow is best because of Strength of the Grave. Best if you take level 1 as a Sorc to get Charisma save proficiency.
6. 15+ Charisma, 1-4 levels: Undead Warlock. Bonus action to go into form of dread and frighten enemies. You need a good Charisma to make it work effectively though. Downside is no shield and no absorb elements.
7. 13+ Strength, 15+ Charisma 4 levels: Vengence Paladin. Paladin is a Gish right out of the box. 15 Charisma lets you use smite spells.
8. 15+ Wisdom and 7+ levels: Fey Wanderer Ranger. Is 7 levels really a dip? This subclas is a straight Gish even if you single class, 7 levels lets you firghten or charm with a reaction any time someone makes a save. Pick up Shadow Touched with Cause Fear and go to town.
9. 13 Intelligence 4+ levels: Armorer Artificer. Lets you use heavy armor. Magic blasing from your armor. Would be higher if he had shield spell.
What subclass and how many levels?
I would go with bladesinger. As far as levels go it depends on how much magic you want keeping in mind that the more levels in BM you take the slower your spell progression is.
It depends on what options you want, how you want to play it and what your abilities are. Here are a few options:
Options:
1. High Strength and High Wisdom: Heavy armored Temepest Cleric. Either 4 or 6 Battlemaster levels, the rest Cleric. If you go 4 levels pick up Hex, Booming Blade and Green Flame Blade from Magic initiate, alternatively start as a high Elf for Booming Blade. You can also look at gift of the chromatic dragon. This build has only 1 attack a round but it is very Gish and you want to boost booming blade damage to be as high as possible (Hex+Metalic Dragon+Divine Strike). You can maximize Thunder and Lightning damage with CD and move the enemy every time you lightning them. If you take 6 levels in Battlemaster you have multiattack and would use it instead of a cantrip.
2. High Strength and High Charisma: Battlemaster 6, the rest Paladin: Divine Smite, a bunch of smite spells. I like Vengence and Conquest for this. Wrathful smite and shadow touched with Cause Fear if you are conquest. I also prefer a Goblin-Conquest for this build to be able to disengage as a bonus action after you have frightened the enemy.
3. High Dex and High Intelligence: Battlemaster 4, Bladesinger X: Any race works well here. This is the combo that has the most flexibility you can be a very powerful melee character and a very powerful fighter. Again Goblin for disengage combined with Bladesinger Extra Attack and Booming Blade is nice. This iis the combo that has the most flexibility.
4. High Strength and medium Intelligence: Battlemaster X, War Wizard 4: Much more of a fighter than a Wizard but War Mage gives you the spells you want as well as the reaction +4 to saving throws (and +2 tto AC once you run out of shield spells). I like a Variant Human for this build picking up the heavy armor master at 1st level.
5. High Odd Strength and High Charisma: Battlemaster X, Shadow Sorcerer 4: Again more of a Figther than caster. With an odd strength would start as a sorcerer here for the charisma save with Strength of the Grave and then take heavy armor through a feat while boosting strength. Lots of Races work really good here. Damphir to be able to walk on ceilings (get a reach weapon and attack from the ceiling). Shadar Kai to use blessing of the Raven Queen and get resistance to damage. Half Orc which gives you relentless endurance and then the ability to do it again through strength of the grave.
6. High Dex, High Wisdom: Battlemaster 6, Ranger X. Any Ranger subclass works here. Fey Wanderer seems to be the most Gish build to me if you have a high Charisma. Dragonborm with the Dragon Fear feat works particularly well/
It depends on your abilities at 1st level and how many levels you want to devote to a "dip" With that in mind here are the best options in order IMO:
1. High Intelligence (15+) 2+ levels: Bladesinger is the best dip assuming you go at least 2 levels. This character can go anywhere from 2-20 and will outmatch any other option at any level. The only downside with a Bladesinger is you are actrually worse because of your fighter levels. A straight non-multiclassed bladesinger who selects spells specifically for melee is going to be a better Gish than a dex-based Bladesinger-Battlemaster. So it is basically when do you want to jump off.
2. 13-14 intelligence 2 levels: War Magic Wizard is awesome as a multiclass Gish. If you have a high intelligence Bladesinger is better. If you have a low intelligence you can't be a Wizard, but if you have a 13-14 a 2-level War Magic dip is awesome. +1 or +2 to initiative, +4 to saves, +2 to AC once you run out of shields.
3. 13-14 Intelligence 2 levels: Diviner gives you the spells you want and portent. This is not as good as War Magic though.
4. 13+ Wisdom: 1-2 levels: Cleric. Any Cleric will work for a dex character but Arcana and War Magic lean into Gish the best IMO.
5. 13+ Charisma, 4 levels: Sorcerer. Any Sorcerer would work, Shadow is best because of Strength of the Grave. Best if you take level 1 as a Sorc to get Charisma save proficiency.
6. 15+ Charisma, 1-4 levels: Undead Warlock. Bonus action to go into form of dread and frighten enemies. You need a good Charisma to make it work effectively though. Downside is no shield and no absorb elements.
7. 13+ Strength, 15+ Charisma 4 levels: Vengence Paladin. Paladin is a Gish right out of the box. 15 Charisma lets you use smite spells.
8. 15+ Wisdom and 7+ levels: Fey Wanderer Ranger. Is 7 levels really a dip? This subclas is a straight Gish even if you single class, 7 levels lets you firghten or charm with a reaction any time someone makes a save. Pick up Shadow Touched with Cause Fear and go to town.
9. 13 Intelligence 4+ levels: Armorer Artificer. Lets you use heavy armor. Magic blasing from your armor. Would be higher if he had shield spell.
Gloomstalker 3. Amazing spells from Ranger and amazing features that synergize amazingly well. Action Surge + Dread Ambusher is broken.