There are three different types of d20 rolls in 5e - one of them has the answer you're looking for.
Ability checks. These don't come up in combat much - only in contested ability checks for grappling, shoving, and hiding. Both you and your opponent roll a d20, maybe add something to it, and compare to see who rolled higher. Nope.
Attack rolls. You know these - you make a weapon or spell attack, roll a d20, add some stuff to it, compare the number to the target's AC, and see if you hit or miss. You don't like these.
Saving throws. These are used for many spells - you just declare you're casting so-and-so spell, and you're done - the onus is on the target(s) to defend against it. You don't roll any dice!
If you are new to DnD or 5e, some quick info: saving throw spells most commonly do a couple of things: impose debuffs on either one or several enemies with a Constitution or Wisdom saving throw (see Blindness/Deafness and Slow), or deal area-of-effect damage and prompt enemies to make a Dexterity saving throw - and often still take half damage if they succeed. (see Fireball) There are others, like single-target spells with Dexterity saves (see Disintegrate) and occasional spells that use Strength, Charisma, and Intelligence saving throws (see Gust of Wind, Divine Word, and Phantasmal Killer) but those are the most common.
Just keep in mind that saving throw spells are generally best at debuffing/disabling and area damage/minion clearing. To build a character around these spells, I would suggest a Bard, Sorcerer, or if you want to be on the front line, a Cleric.
Bards get the classic saving throw cantrip, Vicious Mockery. It doesn't do a whole lot of damage, but it really reduces the effectiveness of big single targets. Bards get a fair number of saving throw spells, focusing more on debuffs and disables but also getting some AoE damage, and they also get to steal spells of your choice from other classes. They also make good party healers, although that will require you to roll dice.
Sorcerers get relatively few spells, but also get Metamagic, which makes them good for focused builds like this. They have a lot more innate access to AoE damage, but a little less to debuffs and disables. In particular, their Metamagic option Heightened Spell is great for augmenting their saving throw spells, giving a target disadvantage on the save. If you're playing with the new spells from Elemental Evil, take the Frostbite cantrip, which works much like Vicious Mockery.
Clerics are the only option that get easy access to medium armor, heavy armor and shields. Their main combat cantrip, Sacred Flame, is a saving throw, and you can augment it by picking a class that has the Potent Cantrip feature, allowing you to add your Wis modifier to its damage. If you want to buff the party, Clerics have the amazing spell Bless, and at level 5 they get access to the godly spell Spiritual Guardians, which does about half the damage of frickin' Fireball in a 15-foot area around you every round! Later on, they get access to another incredibly powerful Cleric-only spell, Contagion. Like Bards, Clerics also make great party healers.
I hope all that helps you find a character you enjoy playing!