Should I do a full test of all the water after it fills before adding anything?
That's always best. Once you confirm your new fresh-fill numbers, start adjusting. There's no need to hold on adjusting other chemicals such as FC/CYA, or CH. Many can be done simultaneously except for chlorine and muriatic acid. Keep those separated by a a few minutes. Below is from the Spa Sticky.
Water Balance
If your CH is much below 100 ppm, raise it to ~150 ppm. Otherwise, make a note of it.
Adjust your TA to ~50 ppm. Use Baking Soda to raise it (unlikely to be needed), or Acid/Aeration method to lower it. It takes ~8 oz Dry Acid to lower TA by 100 ppm in a 350 Gal Tub. Don't add it all at once!
pH should be between 7.2-7.8. Aerate to Raise pH. Acid to lower it.
If pH creeps up too much (>8.0), lower TA. If pH is too low, and doesn't raise enough by aeration, raise TA.
Add ~50 ppm Borate. (Borox/Acid or Gentle Spa)
Calcium, Baking Soda, Dry Acid and Borates is all you need to balance your water.
Use
PoolMath to calculate exact amounts to add.
Sanitation
On a fresh fill use Dichlor until you get to 20-30 ppm CYA. (10 ppm FC = 9 ppm CYA)
Then switch to plain bleach or liquid chlorine.
Never let FC drop to Zero for any length of time. Keep it between 3-6 ppm normally, min of 1 ppm, and shock to 12 ppm once a week.
Use MPS if needed before hot tub party's and/or once a week to help oxidation. Don't use more than needed, because it's acidic and may lower your TA/pH.
Dichlor, Bleach and MPS is all you need to keep your tub sanitized.
Use
PoolMath to calculate exact amounts to add.