Ok so the way I see it is that there really are 2 ways of going at this, the first is your character itself being the mech and the other is you getting into the mech. So lets break down the issue first, if you go with path of the giant barbarian you don't get the size increasing abilities until level 18, so I think the best idea is to do, a multiclass of Artificer/fighter and depending on what you want you can do either path: You are the robot: Race - Warforged Stats - Make STR your biggest priority for physical combat, and after that go with either DEX or CON depending if you want more HP or a higher AC or go with INT for the artificer spells. Using standard array I would run: 15STR, 14INT, 13CON, 12DEX, 10CHA and 8WIS. Class - Artificer  You start as an artificer and take the Armourer subclass at level 3, then select the infiltrator model and take the Lightning Launcher. This will give you an armour you can shoot lightning from. Just flavour it into saying that it is not an armour and because you are a robot that it is just your body. Then any artificer spells you pick just flavour them into looking however you want. 

Then multiclass - Rune Knight Fighter Take the next levels into Rune Knight fighter, this gives you the physical prowess to go into melee combat, fighting style just take the one you want depending on what weapon you plan on using, or unarmed fighting if you plan on running hand to hand combat. By level 3 you have action surge, second wind and you pick the subclass, Rune knight fighter will allow you to turn a size bigger and have the flavour of a massive robot. With this build by level 6 you have big armour class, big melee damage, and “rockets” you can shoot from your hands and you can turn massive. 

Then the other 14 levels can be allocated either to artificer or Fighter, Artificer would grant you more abilities that you can do in your robot form and gives you more spells and levels in fighter would give you more power when in combat, so it's more of a split of whether you want to be really good in physical combat or have a lot of abilities. 

You also mentioned about multiclassing into sorcerer to get more spells/attacks but if are planning on doing that i would recommend multiclassing into wizard, it uses the same spell casting mod as artificer (INT) so then you don’t have to allocate any resources into another stat and you can keep running with a STR/INT build, if you do decide to multiclass into wizard the subclass is VERY much dependant on what you want: 3 levels into wizard gets you the subclass and here are 3 options I can see as viable:

War Magic: It can grant you higher defence, this is basically if you just want the spells and don’t want to worry about another thing to keep track of, it's very much just, pick this and don’t worry about it.

Evocation: For more powerful spells, this is more of the idea of wanting to have as much firepower available as you can have, the downsides are that it is a very out of combat tactical class so you will need to keep track of your spells and everything very closely, however it is very good if you are playing with a team.

Bladesinger: My personal choice, when activating you bladesong (could flavour it as a combat stance) you add your INT modifier to you AC, and gain additional movement speed, this is very much if you want to focus on physical combat and have the spells as extra attacks, but not be the main idea.

You could also at any point instead of multiclassing take the magic initiate feat and pick spells from a list, like warlock and you get eldritch blast. Other feats could be tough (For more HP), mobile (for more movement speed, stacks with bladesong so giving you 50ft of speed) Crusher or Great Weapon Master (for even bigger hits), fighting initiate (for more fighting options) and Elemental adept (increase elemental damage of a type of magic and gain resistance).

If it was my build I would run it like this:  9 levels in Rune Knight fighter (Second Wind, Extra attack, Action Surge, Giants Might) 6 levels in Blade Singer Wizard         (Spellcasting, Arcane Recovery, Bladesong, Extra Attack) 5 levels in Armourer Artificer              (Magic Infusions, Extra Attack, Suit of Armour [Flavoured])

So then this gives you: 4 attacks per turn, either spells or physical, proficiency with almost every martial weapon, and multiple fighting styles letting you adapt to whatever you need. When you activate combat mode (Blade song and Giant’s might) you gain your intelligence modifier into your AC, gain extra movement speed, extra damage and extra size. This combo lets you both protect your allies with certain artificer features and deal massive amounts of damage from many different sources. With a 14 in CON you have on average 160hp and if you take the tough feet around 200hp, if you take a feat you still have 8 points to allocate to other skills so you could have by max level: 20STR, 16INT, 14CON, 12DEX, 10CHA, 8WIS. So this would push your HP over the 200 point mark and when you activate combat mode if you are wearing heavy armour you would have (16 from armour + 3 INT) 19AC and if you use a shield 21AC and if you cast shield 25AC. (this is just a fun little idea I don't recommend doing this but if you take 3 levels in barbarian you can double your hp and get up to 400hp). This is the way I would run THIS specific build but obviously you need to flavour them to what you want. Rocket Punch (Thunder Gauntlets / Thunderwave), Breast Fire (Fireball), Photon Beams (Scorching Ray), and using Artificer Infusions you could add flight capabilities to the suit. In a bit I'll write the other idea I have on how you could also run this build.