Easy.
You know that there will be no CYA, so you can use poolmath to calculate how much and add that at any time it's convenient. But soon. Take the time to read this How-to:
Pool School - PoolMath
You know there will be very little FC in tap water, and the sun will destroy it all within an hour until you get some CYA in there, so add enough bleach to get to 3 FC. Add that much every day, actually, until you get CYA in the water and it registers and the FC starts holding.
You have a vinyl liner so low CH is not a problem. And if you have high CH, there's nothing you can do about it since the refill water would be exactly the same as what you just put in, so ignore it. You can test it just for practice, but don't bother doing anything with it.
TA is the last thing you deal with. But you do need to know its value to calculate pH adjustments, so test it.
Which leaves pH. Test that, and if it's above 7.8 or below 7.2, adjust it.
You can add the salt any time. Ideally, you wait until the water is swimmable (pH, FC, temperature) and then you throw a few kids in to help mix it up. They'll have a blast.
When you've got the pH dialed in, the CYA dissolved, it's holding FC, and the salt is dissolved for 24 hours, turn on the SWG and let it take over chlorinating duties.
If you have a decent test kit, you should be able to test, make a shopping list, make one trip, and deal with it all and have it all set for the summer before the end of the weekend. Maybe as soon as tomorrow night!
If this all reads like Greek, then test the water and post results and we can walk you through a step at a time with specific answers.
looks like less than a gallon is needed! Yay!