So, 5e mostly does a great job of representing the gish, spellblade, swordmage, etc. almost every class can do it to some degree without multiclassing, and any option that lets you pick up a cantrip can get ya part of the way there. 

That said, there are gaps, and I've two different homebrewed settings where "gish" style fighting is pretty common, with many different schools and traditions to draw from. World's where a purely non magical character is probably less common than a character with some minor magics. 

So I thought, why not have some more options that aren't subclasses do gishing? 

First, identify the remaining gaps. 

• Too few weapon cantrips
• little support outside of those spells and some class features for slinging spells and using a sword
• no specific feat support
• many classes don't go far enough with subclass features in supporting gish characters actually gishing

cantrips

Note: I'm not going to write it all out right now. Assume that the basic info is he same as booming Blade or green Flame Blade.

Burning Blade: If the target fails the saving throw, they are on Fire, and take 1d4 Fire Damage at the start of every turn until they put the fire out by making a successful Dex Save. If they do so as an Action, they automatically succeed on the save. 

Add 1d4 at appropriate levels 

Touch of Frost: failed save= their movement is slowed by 10ft, str save every round to break the ice and End the effect. Can use whole movement for the round to end effect without save. They take 1d6 Cold damage when the effect ends. 

Damage by level as normal

 What about one that blinds the target? 

From another angle, what about a shielding cantrip of some kind? Seems like the SCAG cantrips have the power level of dual wielding with the fighting style, so what is a good magical equivalent to the defensive and protection fighting styles? 

What about mobility? Ability to jump unnaturally far and make an attack as part of or at the end of the movement? Short teleport and attack? 

Perhaos one that simply does a variable type of damage? Like it can do fire, cold, poison, necrotic, etc? 

Life drain? You hit them and gain HP, THP? 

HEALING PIÑATA? lol you hit them and the next attack against them heals a small amount? 

fighting styles

Spell and Sword: While wielding a one handed weapon or fighting unarmed, and neither wielding nor holding anything in your other hand, [what?]

• Use a mental stat for attack and damage? 
• Add Str or Dex to damage with cantrips? 
• Make a weapon attack with no stat bonus to damage as a bonus action when you make a spell attack? 
• Add int, wis, or Cha mod damage of a type chosen when you take this? 

Spell Shield: As above, but defensive.

• You gain a +2 to AC? 
• Can use a reaction to add proficiency bonus to a saving throw against a magical effect?
• Reaction to knock a creature back 5ft when it damages you with a melee attack? 
• +1 to saves against spells? 
• Use a reaction to absorb magical damage (equal to spellcasting stat mod?) and either turn it into THP or add it to next attack's damage?  

Spell Blade Dance: Something more like Mariner, with utility. Minor Teleportation? Add spellcasting stat mod to speed, acrobatics and distance of jumps? Double jump? Fly speed, but you have to land every turn? 

Feats

leveled spells

Miscellaneous

If it were me, I would not go in the direction of Fighting Styles especially since the Fighter wouldn't really benefit from it until 3rd level when they become an Eldritch Knight and it would be difficult to design without stepping on the toes of the Eldritch Knight's features. Just wanted to get that out of the way first.

While I get the idea behind Burning Blade it feels too close to the 1st level spell Searing Smite. Granted, not every class has access to that spell, but I thought it was something to keep in mind. With regards to Touch of Frost, I like the concept but here's how I'd handle it:

Rime Blade
Evocation cantrip (sorcerer, warlock, and wizard)
Casting Time: 1 action
Range: 5 feet
Components: V, M (a weapon)
Duration: 1 round
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spells range, otherwise the spell fails. On a hit, the target suffers the attack's normal effects, and is wreathed in frost until the start of your next turn, halving the target's speed. On its turn, the target may use its action to cause the spell to end. When the spell ends, the target immediately takes 1d6 cold damage.
   This spell's damage increases when you reach higher levels. At 5th level, the melee attack deals an extra 1d6 cold damage to the target, and the damage the target takes when the spell ends increases to 2d6. Both damage rolls increase by 1d6 at 11th level and 17th level.

For the healing one, there's actually a fitting candidate from 4E's Divine Power. Here's my attempt at converting it to 5E.

Recovery Strike
Evocation cantrip (cleric)
Casting Time: 1 action
Range: 5 feet
Components: V, M (a weapon)
Duration: 1 round
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spells range, otherwise the spell fails. On a hit, the target suffers the attack's normal effects, and the first time an ally hits the target before the start of your next turn, that ally regains a number of hit points equal to your spellcasting ability modifier, and the spell ends.
   This spell lasts longer when you reach higher levels. At 5th level, the spell only ends after the second time an ally hits the target. At 11th level, the spell only ends after the third time an ally hits the target. At 17th level, the spell only ends after the fourth time an ally hits the target.

Excellent! When next I get on a computer I will update with those, and give you credit. 

Fighting Styles: good points, but that is only one class. Some of them definately would be weird with a class that gets FS before spells, though. Maybe those just aren't available to the fighter? 

Rime blade gains 2d6 at the appropriate levels, far more than any other cantrip.  Overpowered?  I think so. 

thats the same progression as the cantrip its based on