The important question is which part matters to you more - being physically strong and swinging a greataxe, or burning everything to the ground with fire magic?
AFAIK we still don't know the details about how multiclassing will work, but I think there's a good chance you'll get the most enjoyment out of your character if you make a choice. Nonetheless, I'm sure a hybrid build can be effective and fun too if you like that flavour, although a bit more complicated to build. Here's some options:
Barbarian (Berserker)
Pro: If you want to cleave things for massive damage with a big axe, this is your easiest option. Barbarian has the highest pool of hit points and the highest potential per-hit damage with a two-hander. It also has Unarmored Defense - Armor Class equal to 10 + DEX + CON - which evokes the feeling that you need no armor to be defensible and can only rely on your scales for protection. STR is your main ability which determines your melee damage and chance to hit, followed by CON which increases your already massive HP as well as AC. DEX is useful for extra AC and you can safely drop WIS and INT to put the remainder of your points into CHA. 12 CHA and Intimidation proficiency is enough for you to be reasonably good at menacing people.
Con: As a downside, this class brings no magic to the table and your only source of fire will be your meagre dragon breath or possibly a flaming weapon.
Sorcerer (Draconic bloodline)
Pro: A Draconic Sorcerer inherits their power from a dragon ancestor, which I think is a fantastic thematic fit for a dragonborn who fancies themself a true dragon. The Sorcerer is a pure spellcaster. You gain additional damage with fire spells if you choose the right bloodline, access to devastating spells like Fireball and dragon scales that protect you like armor. Draconic Sorcerers are also surprisingly durable compared to Wizards (or even other Sorcerers) thanks to their permanent AC equal to 13 + DEX, +1 HP per level and proficiency in CON saves. CHA is your main ability that affects the power of your spells, followed by CON and DEX which significantly improve your survivability. CHA being your highest stat also sets you up to pass even difficult Intimidation checks, so ironically you'd be more terrifying this way than as a Barbarian.
From 3rd level onward the Quickened Spell metamagic allows you to cast two spells per turn (unless this rule is changed on release). If you haste yourself to gain an extra action, you can open a fight with 3 fireballs which is something that not even an evoker Wizard can do. I can promise you one thing: it doesn't get more "burn everything to the ground" than this.
Con: The obvious downside here is you won't be swinging any big weapons and you'll be best off avoiding the front line.
Fighter (Battlemaster) 5, Sorcerer (Draconic) 7
Pro: Here's a way to be a towering juggernaut who can swing a massive weapon and throw a fireball or two as needed. You begin as a Fighter, which grants you proficiency with heavy armors and all weapons. STR and CHA should be your highest stats, followed by CON. At level 5 you gain Extra Attack (two attacks per turn instead of one) and at this point you can begin leveling in Sorcerer. By the end of the game you'll be marching into battle in full plate armour, swinging your giant sword twice and dropping a 4th level fireball on people's heads as a bonus action, all in a single turn.
Con: The problem is, it takes a while to get there. The first 4 Sorcerer levels won't do much for you, but they can still be very useful if you pick the right spells. For example, Shield is a fantastic reaction for melee characters and Scorching Ray provides a fiery ranged attack. Another problem is you'll be missing out on one ability increase/feat, which you'd normally gain at levels 4, 8 and 12. You'd best start with a 17 in both STR and CHA so that you can round them up in one go. At the end of the day, you may not be as good at either martial combat or spellcasting as a dedicated character, but you can absolutely have an effective and strong character that feels like a mesh between the two and is fun to play.
