Upgrade Calibur first because it is the best weapon against all types of enemies. You should get Polaris because the expanding bullet is just a joy to use (not if you are with a lot of people) and the knockback can save you from enemies.
What Should I Level First? Calibur vs Sealed Sword (Avenger) / Pulsar (Polaris) vs Voltech Alchemer
Calibur is a bad weapon which annoys teammates, get a piercing or shadow weapon to go alongside your divine avenger... That way you're either strong or normal against all enemy types :D
I'd suggest:
Piercing you have flourish or snarble barb (basically the same but with different pecials)
Shadow you have Faust or nightblade (if you have an avenger you'll probably come to prefer faust out of the two)
Leviathan is the best sword in this game, it is the only sword in which you don't have to hunt for the recipe and is able to kill the Jelly King, The twins, and Vanaduke. The charge attack is basically a mini-nitronome bomb that causes victims (aka your teammates) to hate it because their heads end up getting smashed by a monster you just blasted (blasting teammates can be avoided).
Saying calibur is bad because it annoys teammates is bad as the argument that shivermist is bad because it traps your teammates or polaris is bad because the teammates can't see the enemy. Don't blame the weapon, blame the user.
I suggest getting leviathan as soon as you can, then your armor and then upgrading the rest. With a leviathan you won't have problems dealing with mobs where as having divine avenger (a heavy weapon) is harder to do so. Generally, weapons matter more than armor because eventually you'll have to learn to dodge instead of relying on your shield or armor. In this game, DPS is king which is why people often choose Pulsar over Catalyzer.
The damage increase from 4 to 5* weapons is minimal and only really noticeable in the final stratum. On the other hand, armors offer a great deal more defense when going from 4 to 5*, particularly Vog Cub which receives elemental defense, fire resistance and Sword ASI. Upgrading your armor is going to increase your survivability.
I would suggest upgrading your armor to 4*, then your weapons to 4*, then your armor/shield to 5*. Finally upgrade your weapons to 5* in the order that you use the most.
Get DA first, then Vog cub, then shield, then storm driver, then polaris.
If you have a Polaris, don't bother with Voltech. Get your weapons to 4* first, depending on which you use most or anticipate using most.
Honestly is it that hard to decide for yourself in what order to heat/craft? Okay, just craft a Vog Cap and leave all your weapons at 2*/3* forever.