Here are (at least some) of the ways you can avoid being affected by the frightful presence. I've excluded features that give bonuses to saving throws, as I'm not sure how much it will actually benefit you if your mod is only +0, and I excluded anything that gives advantage against fear, because advantage cannot raise your maximum roll.
The Monk's Stillness of Mind (PHB 79)
Starting at 7th level, you can use your action to end one effect on yourself that is causing you to be charmed or frightened.
The Beserker's mindless rage feature (PHB 49)
Beginning at 6th level, you can’t be charmed or frightened while raging. Ifyou are charmed or frightened when you enter your rage, the effect is suspended for the duration of the rage.
The Paladin's Aura of Courage (PHB 85)
Starting at 10th level, you and friendly creatures within 10 feet of you can’t be frightened while you are conscious.
This feature is probably the best one, since it doesn't cost any resources at all and it protects your friends as well. Thus, the Paladin is the best class for resisting fear, second to the Barbarian (as one might expect).
Power Word Heal (PHB 266)
If the creature is charmed, frightened, paralyzed, or stunned, the condition ends.
Calm Emotions (PHB 221)
You can suppress any effect causing a target to be charmed or frightened. When this spell ends, any suppressed effect resumes, provided that its duration has not expired in the meantime.
The Heroes' Feast spell (PHB 250)
A creature that partakes of the feast gains several benefits. The creature is cured of all diseases and poison, becomes immune to poison and being frightened, and makes all Wisdom saving throws with advantage.
Heroism (PHB 250)
Until the spell ends, the creature is immune to being frightened and gains temporary hit points equal to your spellcasting ability modifier at the start of each of its turns.
Hallow (PHB 249)
Second, you can bind an extra effect to the area. ...
Courage. Affected creatures can’t be frightened while in the area.
Blackrazor (DMG 216-7)
While you hold the weapon. you are aware of the presence of Tiny or larger creatures within 60 feet of you that aren't constructs or undead. You also can't be charmed or frightened.