So my Copper Dragon born Barbarian just hit level 6.   I use a Great Sword +1 in battles and run the Path of the Berserker for a primal path choice. 

I've just acquired an Aberrant Dragon mark which grants me a cantrip and a 1st level Sorcerer spell. My dilemma is I don't know which cantrip and spell I should choose to up my melee game.  Right now I'm leaning towards "Lightning Lure" for my cantrip & "Expeditious Retreat" for my spell. Originally I was going to go with "Sword Burst" & " Acid Stream" for being in the thick of battles but as I stated i'm still unsure what would work best.

I'd love some feedback as to what people think would make my Barbarian more barbaric

Just remember that you cannot cast spells while raging.

I guess the obvious question is; what do you feel your character needs?

As Jounichi1983 says, you can't cast spells while raging, and rage also ends concentration, so you probably don't want any concentration spells. While you can technically still cast other spells and leave them running as you rage, there aren't a lot of choices in the Sorcerer list at 1st level.

So with this in mind, what do you envision your Barbarian actually using spellcasting for? There are lots of good attack spells and cantrips for Sorcerers, which you could certainly use before raging, or as a ranged option when you're about to lose Rage anyway by being out of range to attack normally.

Charm Person is an interesting option both out of combat and just before, as you can potentially take someone out of the fight, while out of combat you can use it to get past guards and such. Though if your spell save DC is likely to be low I'm not sure if it's worth it.

The one I'd go for personally is Disguise Self; it's a super useful social stealth spell as it lets you just walk into fancy parties and restricted areas as if you were someone else (anonymous or known); you'll know better than me though if your campaign is one where you're likely to have opportunities to use that, though often I've found it can be up to you as a player what opportunities you create with your party.

For cantrips there are a few good utility options for sorcerers; Mage Hand is one of the easiest to use for thieving, reaching things and for creative shenanigans. Minor Illusion is good for distractions and a few other things, you can also get a lot of mileage out of [Tooltip Not Found] with some creativity (I've used it to clean up blood, clean off clothes etc., but I mostly use it to mess about with people's food and drink 😄). Message is useful for stealth (social or otherwise) as you can communicate silently with your party to warn them or get status updates for when they need you to charge in and wreck stuff. Light could be useful as you don't normally have darkvision as a Dragonborn.

So yeah, the first choice is going to be combat or utility; personally I'd lean for the latter, but you can still use combat spells and cantrips so long as you keep in mind how they conflict with Rage, i.e- you need to either cast them at the start of combat, or forego Rage in combats where you think some elemental magic might be more useful. Utility choices avoid that conflict, and will give you a boost to out of combat utility in social settings, exploration and such, but some campaigns lean combat heavy. It really depends what you want for your character.

Hope some of that is useful!

For combat utility false life may be your best bet as it can be cast before battle and requires no concentration.  Don't undervalue out of combat utility spells such as charm person and disguise self either.

Quote from waddingsly >>

For combat utility false life may be your best bet as it can be cast before battle and requires no concentration.  Don't undervalue out of combat utility spells such as charm person and disguise self either.

I was wondering about False Life actually, though as part of the Aberrant Dragonmark you can roll a Hit Dice for a 50/50 chance of gaining temporary hitpoints anyway when you cast your 1st level spell through it (though the other 50/50 is a creature near you takes force damage so it's best used with an enemy nearby). Also since this will be the only spellcasting available, 1st level False Life won't scale beyond 1d4+5 so it's not going to be a huge boost for a Barbarian.

I ran into a similar conundrum with my Bard taking Magic Initiate (Warlock); originally I was going to take Armor of Agathys as my spell choice but it would have been limited to 1st level only, so in the end I took something else and took Armor of Agathys later as a magical secret so I could scale it up and keep it useful at higher levels.

I don't usually Rage unless my party becomes outnumbered or a member has been knocked unconscious from an encounter. I think my needs are more for combat as I'm more melee oriented and the rest of my party more effective with ranged attacks.    I've decide to go with Lightning Lure and Acid Stream .

The LL to either pull attackers away from our casters or healers and into range for my Acid Stream attack or for a beat down with my Greatsword +1

Quote from W00DS1E >>

I don't usually Rage unless my party becomes outnumbered or a member has been knocked unconscious from an encounter. I think my needs are more for combat as I'm more melee oriented and the rest of my party more effective with ranged attacks.    I've decide to go with Lightning Lure and Acid Stream .

The LL to either pull attackers away from our casters or healers and into range for my Acid Stream attack or for a beat down with my Greatsword +1

 You could also consider Booming Blade to discourage or punish people running past you. Lightning Lure is a decent option and will allow for you to help casters to "disengage". A couple of 1sts that could be interesting are Absorb Elements and Earth Tremor. In order for Earth Tremor to be worth it, you'd want as many people in it as possible or just be willing to have the difficult terrain, as there is no damage on a save. 1d6 damage is alright, and proning the fails on top of that works great with crowd control and advantage for melee allies. It's less effective if you are the only melee character unless you're using it strictly for crowd control. 

I know you've already decided on the spells you want, but I just wanted to comment on what I would go for in your situation.

For Cantrip, I would take Mold Earth. It's a great utility spell, but flavor-wise it can be a ton of fun to have this big burly character who can just shift huge blocks of earth at will. Sort of like an Earth Bender.

As for the 1st level spell... even though it's probably not the most useful, I'd go with Absorb Elements. You probably won't use it every day, but it would be so satisfying if you get hit while not Raging, absorb the damage, then turn around and smack the guy with an extra d6 of energy stacked on top.

Shield would also be a good 1st level spell that could fit a barbarian, which won’t mess with your action economy, and only spend your reaction.

I could see thorn whip being good against retreating opponents.

Quote from W00DS1E >>

I don't usually Rage unless my party becomes outnumbered or a member has been knocked unconscious from an encounter. I think my needs are more for combat as I'm more melee oriented and the rest of my party more effective with ranged attacks.    I've decide to go with Lightning Lure and Acid Stream .

The LL to either pull attackers away from our casters or healers and into range for my Acid Stream attack or for a beat down with my Greatsword +1

 Acid stream is unearthed arcana, so your DM may not allow it.