What Weapon Should I Upgrade First?

Depends on which bot you use the most often. Pre update, I had max out tarans and mags since I used the Galahad quite a bit. Now that rockets have gotten a boost might as well do those first if it's all the same to you--more 'bang' for your buck and all that.

Going by what you have posted:
- Mag up lvl 6-7 to match Tarans
- Pinatas 
- Orkans

I leveled everything to 8/10, then went more by bot / use.  I continued to level my brawler bots & weps, then ambush (Griffins).  I always think bots to L9, get max speed and leveling weapons before  bors

It appears you do have any mid-range?  Also, you have 3 Griffins in your hangar.   I run 3 - db, Stuka, and now rdb.  If you have any Tulus & Pins around, you might try them out on the Griffin, they work well.

I've been working on the best upgrade strategy, and while it's far from the finish, I can get you some tips:

DISCLAIMER: I presume you are NOT paying Gold for speeding up, have 5 slot hangar and highest level of stuff in your hangar is lvl10 (it's fine if it is a lot less; my tips just won't work for maxed out hangar well)

1) Aim for no higher than lvl9 to get the best money\time\result combo. This is due to the following facts:
robot speed, obviously;
with each upgrade step, you are getting approximately 10% increase of a firepower. So basically you will get same effect upgrading 10 weapons from lvl 1 to lvl 2 or just by upgrading single gun lvl1 - lvl9.
And you will get 10 weapons upgraded to lvl2 A LOT faster and cheaper. Let's say you got a Magnum Patton, so you can either upgrade each weapon from lvl1 to 2, and you will get +10% firepower for a bot for 40,000 Ag and 20 minutes, or you can upgrade single weapon to lvl4, get +11.6% bot damage for 70,000 Ag and 1 hour and 35 minutes.
So it's better to have every weapon you use often at +\- similar level

2) However, if you use some weird combos like Trident+Pin Natasha - you will be better off upgrading Tridents first, much higher. Because your Pins are doing just a tad of your bot total damage.

3) Upgrade weapons first, robots later (with discrepancy of about 1-2 levels between bots and weapons) - low level weapon deal less damage, in % of a robot HP, than high-level weapon - example is that on lvl1 Destrier(44k) would be killed by lvl1 Trebuchet (6.2K) in 8 shots, but lvl12 Destrier (87k) would be destroyed by lvl12 Treb(17.5K) in just 5 shots. So usually it's a good idea to upgrade weapons first (with notable exceptions of a beacon runners)

4) So as a general rule, I recommend taking EVERYTHING you use to level4 first, then do a round of weapon upgrades to lvl5, then let bot levels catch up, then get to lvl6, and repeat until everything is at lvl9

5) Plan for expenditure upfront! Upgrade expenses tend to raise +100% each level. So while Magnum costs laughable 10,000 for lvl1-lvl2 upgrade, it would be a whole lot of 13,000,000 for lvl11-12 upgrade. So plan for it!

6) In continuation of p.5 - you don't want to let your upgrade process be idle. So if you see your Silver is running low - it is better to switch to upgrading some non-priority low-level stuff you plan on using "sometime later". It is way better to spend a week or two upgrading some stuff from the attic, meanwhile grinding for silver, than to blow your money on something cool and let the time to be wasted afterward.

7) Remember, many items are pre-upgraded to some level right off the store (think: heavy bots, weapons, some gold stuff, too) - so in order to be able to compare stuff you need to take some low-level stuff to an adequate level. Here comes my p.4 (above) in handy.

P.s. and don't forget to check out WIKI to see what weapon should, and what shouldn't be upgraded (spoiler alert: you should put your Punisher to the trash and replace it with Magnums after 2.9 update!)
warrobots.wikia.com/

Oops, sorry, I didn't see this.
Anyway, I would say your Carnage is absolutely magnificent right now so leave that alone, Pinatas and Trans become very deadly after a while, that's more than I can say for magnums (unless you're doing a quad magnum Patton), I voted aphids, even just 2 aphids at level 10 become annoying to dangerous. 

If you have upgrades to do, and orkan is one of the considerations, it's always orkan.

That is one sound decision.
Just try to keep your weapons 1-2 level above your bots, you will get more leverage on a field (unless it's the small bots or a Carnage).