What suggestion would you guys have for buffing and debuffing classes and not just healing?

If the answer is a primary caster which sub-classes and spell picks?

I tend to play martial characters so I'm not well versed in the primary casters.

Wizard is very good for buffs and ok for debuffs:

Haste is great for casting on any martial class like Barbarians, Fighters, Rangers and Paladins. 

Slow is great for debuffing enemies to make them easier targets, stop multiattacks etc.

Enlarge/Reduce is good for either: enlarge an ally so they deal more damage for martial casters, reduce allies like rogues so they hide better or any Gnome: they become light enough for you to fly them around with Mage Hand - useful for yourself. Enlarge enemies to make them bigger targets or too big to chase you through small spaces or reduce them to make their attacks weaker.

-- I might come back to add more when I have time. Wizards get a lot of options.

@mikaos Most of the casters have great support capacity. If you have a preference thematically then search by Class.

If you're wanting to dive in mechanically I would encourage you to search through the Spell catalog.

You can search spells by "Tag" and you'll find it very easy to narrow your search and find what you are looking for.

Even cantrips can give useful debuffs  Chill Touch prevents healing. Frostbite gives disadvantage for an attack. Ray of Frost reduces movement.

Other spells, not all from the Wizard list: Charm Person, Earth Tremor, Fog Cloud, Levitate, Grease, Cause Fear, Stinking Cloud, Faerie Fire, many many more

Buffs Shield of Faith, Bless, Enhance Ability, Invisibility, Magic Weapon, Pass Without Trace,....

I think cleric is the optimal choice, if just because you have the whole spell selection available to you every time & never have to commit to a particular direction.

Debuffs:

• Bane (1) is one of the most popular in the game, as it is one of extremely few ways to debuff enemy saving throws - ideal for other casters, monks or anyone else who forces the enemy to save. On top of which it is a credible general debuff simply for hindering enemy attacks.
• Blindness/Deafness (2) is a debuff that doesn't require concentration and can undermine attackers/caster (respectively) for a minute.
• Hold Person (2) is incredibly dangerous in team settings, though only war domain clerics get access to the Hold Monster (5) counterpart.
• Banishment (4) is great for taking a foe out of the fight for a long while.

Buffs - there are countless if you want to use your concentration, but these are the stand-outs you can use while maintaining concentration on debuffs:

• Aid (2) is a temporary hit point buffer  doesn't look great on paper, but it is one people swear by in the long run.
• Death Ward (4) is exactly what it sounds like.

There's a huge number of domain options, but you might want to consider the following:

• Arcana for Teleportation Circle (5), as clerics cannot normally transport the party.
• Grave for the Sentinel at Death's Door feature, which if you watch Critical Role you'll know is incredible & unique to this subclass.
• Tempest if you want to go in an environmental effect duration e.g. mists & blizzards to control the battlefield.
• Trickery for the Pass Without Trace (2) party buff, and Polymorph (4) which is incredible as both buff & debuff.
• War Domain for Hold Monster (5), though it also has a channel divinity reaction that essentially lets you turn any ally's miss into a hit (+10 after seeing roll).

I think bards and druids get a wide range of support spells and even support features.

For martial support options. The first thing that popped into my mind is battle master fighter, but pther classes have subclasses good at support too.

I notice that sorcerer hasn't been mentioned any reason other than the spell list that they get?

Quote from Mikaos >>

I notice that sorcerer hasn't been mentioned any reason other than the spell list that they get?

 I'd say not only the spell list but also the spells known are also a factor. Sorcerer's don't get as many Spells Known per level, so people are less likely to select more than one or two support or battlefield control spells.