I think part of what we need to give you the best answers, OP, is to know exactly when in the diffuse Dragonlance Timeline you want to game.
In the iconic War of the Lance time segment, to my admittedly limited knowledge, you can still roughly divide it into Early War (no Divine Magic) and Late War (Divine Magic). Post-War of the Lance adds even more complications.
Because it's what I know best, I'll focus on the War of the Lance segment of the timeline.
Common Races:
Human
Dwarf: Both Mountain Dwarf and Hill Dwarf are viable options. There is a Dark Dwarf clan that roughly corresponds to the Duergar, but these were isolationists and, I think, aligned to Takhisis during this point in time, so DM discretion applies.
Elf: High Elves (Silvanesti) and Wood Elves (Kargonesti) both exist in Krynn at this point in time. Drow do not exist in Krynn - "Dark Elf" is an elf-exclusive Background that would need to be homebrewed. You could also consider adding homebrewed Dimernesti (Aquatic Elf with Otter shapeshifting) and Dargonesti (Aquatic Elf with Dolphin shapeshifting) abilities, but these were isolationist races and did not contribute overmuch to the events of the War.
Gnome: Use the Rock Gnome stats plus fluff to represent Minoi (Krynnish Tinker Gnomes).
Kender: Either homebrew a new subrace for Halfling, homebrew a new race entirely, or mandate that all "Krynnish Halflings" must be Chaotic aligned Lightfoot Halfling Rogues.
Half-Elf: Rare, but still a noted member of the "common" races of Krynn.
Forbidden Races:
Half-Orcs & Orcs: There are no orcs on Krynn.
Dragonborn: These do not exist on Krynn.
Draconian: At the time of the War of the Lance, these are all slaves to the Dragon Armies. A DM may, at their discretion, allow for defectors or redeemed captives, but this would be very unusual.
Lizardfolk/Kobolds: These races live on a different continent to Ansalon, as far as I'm aware.
Rare Races:
Goblinoid: All three tribes of goblinoid exist on Krynn and have some freedom, but at the time of the War, they are mostly draftees to the Dragon Armies.
Half-Goblin: The Krynnish equivalent to the Half-Orc; flavor-wise and ability-wise it's very different to the Half-Orc, so it would need to be homebrewed. I have a draft of this I've been trying to get critiqued.
Goliath: Use stats from this race to represent Ogres or, perhaps better suited, Half-Ogres. As with Goblinoids, these are mostly slaves to the Dragon Armies at the time of the War of the Lance.
Minotaur: Use the stats from the Waterborne Adventures UA, as it's explicitly based on the Krynnish Minotaur. However, these are slave-soldiers of the Ogres at this time, so whilst they aren't invested in the Dragon Armies, defectors/escapees are rare.
Common Classes:
Barbarian
Fighter: Due to Krynnish lore, DMs may want to ban access to the Eldritch Knight subclass, due to the Towers' simultaneous stranglehold on magic and cultural disdain for "arcane dabblers".
Ranger: You will need to either handwave their spell-like abilities as "feats of knowledge, not magic", use the Non-Caster Ranger, or replace with the Scout subclasses for Fighters & Rogues.
Rogue: As per the Eldritch Knight, the Arcane Trickster may warrant banning.
Wizard: Outcast vs. Member of the Towers of High Sorcery is best handled as a custom Background and the aforementioned Tradition-restrictions (Divination/Abjuration for White Robes, Conjuration/Evocation for Red Robes, Illusion/Necromancy for Black Robes) - membership in the Towers basically amounted to superior specialist bonuses, so they don't warrant a new Tradition. Maybe a Prestige Class, though. Additionally, given their lore of doubling as priests of the Lunar/Magic Gods, Theurges with the right Domains could also work as High Sorcerers.
Banned Classes:
Druid: To my knowledge, these don't exist in Krynn until after the Chaos War, although you could argue them as an alternate form of Nature Cleric once the gods return.
Sorcerer: These don't exist on Krynn until after the Chaos War.
Restricted Classes:
Cleric: These do not exist until the gods return in the latter part of the War of the Lance. You may want to consider replacing them with Favored Soul Sorcerers, given the Favored Soul is analogous to the Mystic and the Mystic would better fit the "spontaneously imbued with power" fluff of the First Returned.
Artificer: If not banned outright, refluffing this as non-magical and restricting it to Tinker Gnomes would exemplify the Minoi's status as mad inventors.
Knights of Solamnia:
From all that I know about these, I see no reason to give these any particular homebrewing content. Lore-wise and mechanics wise, prior to the Gods' Return, they work just fine as Human Fighters (Banneret, Knight or Cavalier) with either the Noble Background (established family) or, perhaps, the Criminal Background (fallen families like the Brightblades). Once the gods return, their return to grace is represented by the ability to start multiclassing as Paladins (Oath of Devotion).