Astarion is a great choice. His complex backstory adds depth to the story. He's a vampire spawn with his own motives and development throughout the game. Another good one is Shadowheart. Her mysterious past and her role within the Sharran cult bring interesting story elements. And Gale, his connection to the weave and his pursuit of magic in a rather dangerous way enriches the overall story experience.
A classic and effective party composition could be a tank, a healer, a ranged damage dealer, and a melee damage dealer. For example, a Fighter as the tank to soak up damage, a Cleric for healing, a Ranger for ranged attacks, and a Rogue for melee damage and sneaking around. This allows for a balanced approach in combat and different skills to be utilized during the story quests.
A party with a Cleric, a Rogue, and a Wizard can be great for the story in BG3. The Cleric can heal and provide support, the Rogue for sneaking and lockpicking, and the Wizard for powerful magic to deal with various challenges in the story.
Fane and Lohse are great party members for the story. Fane has a very interesting backstory related to the undead and the secrets of the world. Lohse has her own struggle with a demon inside her, which adds a lot of depth to the story.
For story, I'd recommend Shadowheart. Her loyalty to Shar and the slow reveal of her true self as the story progresses is fascinating. Then there's Astarion. His journey from being a subservient vampire spawn to potentially finding freedom is a great story arc. Karlach is also a good pick. Her bubbly personality combined with her tragic backstory of being in Avernus makes for a very engaging story element in the party.
A composition with a Ranger, a Cleric, a Monk, and a Warlock. The Ranger can be a ranged attacker and tracker, good for exploring and dealing damage from afar. The Cleric is essential for healing and also has some buffs and debuffs. The Monk is a unique melee combatant with high mobility and unarmed attacks. The Warlock has some interesting spellcasting abilities and can make pacts for different powers, which adds a lot of variety to the party's capabilities in the story.
I'd say the Nameless One along with Morte and Nordom. The Nameless One is the main character. Morte is always there to crack some cynical jokes and give insights into the Nameless One's past misdeeds. Nordom, being a modron, offers a very different view of the planes. His logical and somewhat robotic nature contrasts well with the other characters and his abilities can be crucial in solving some of the more mechanical - related puzzles in the game.
For me, Lohse is a must - have. Her internal battle with the demon not only affects her character development but also has implications for the overall story. And Fane, his knowledge of the ancient times is crucial for understanding the world's secrets in the story.
A solid party for the story would be the Inquisitor, Cullen, and Josephine. Cullen has his own story arc related to the Templar Order and his struggles, which ties into the overall story. Josephine is a diplomat, and her skills in handling political and diplomatic matters are very useful for the story. The Inquisitor in this group can make decisions based on their advice and abilities, and together they can drive the story forward.
The best party for BG3's story could be a Barbarian, a Druid, and a Monk. The Barbarian is great for raw combat power and can soak up damage. The Druid can transform and has a variety of spells for different scenarios. The Monk is fast and has unique combat abilities that can help in battles throughout the story.