When you don't know what you're doing it is always a struggle.
Likely you didn't have to take the whole thing apart to thread a new cable but again when you don't know what you are doing it is always much harder.
Generally all you have to do, as has been said, is to release the spring tension of the shifter so the cable port is visible...sometimes you have to remove a top cap or a bung screw/plug to reveal it depending on brand/model...then just insert the new cable.
The soldering thing is a total waste of time and just complicates the whole thing.
Good luck...you are going to need lots of it...

Or.......I can just order a new one. There are two levers, one to release and one to advance. I tried poking each in turn while trying to capture the cut end of the cable. So far no luck.

If the cable you are using is new the end is solid...if it is a used one, or you cut the end off a new one, make sure the strands are nice and tight, if they are frayed...perhaps because a wire cutter was used instead of a cable cutter...you are going to have trouble threading it because just one frayed strand will f it all up.
It doesn't matter if there are two levers or one...you have to release the spring tension to move the cable opening into position.

The big (thumb) lever on each trigger shifter, upshifts, against derailleur spring pressure, that's why it's the big lever, for more leverage. The small (index/trigger finger) lever on each trigger shifter, downshifts, with the derailleur spring helping. (I think) There is also a spring inside the shifter, to make the mechanism downshift when triggered, even if no cable tension; if that doesn't happen, you may need a new shifter. Also be aware, each trigger only shifts one cog or chainring at a time. So to downshift from 3 to 1 on a left triple shifter, you need to pull *and release* the small trigger lever twice, 3 to 2, then 2 to 1. This is known as ratchet shifting, some cars are set up that way, World Rally cars were set up that way with a mechanical ratchet shifter, one yank per shift, until electronic paddle shifters on the steering wheel superseded them. Motorcycles (without automatic transmissions) have been ratchet shifted for over a hundred years.

No need to solder. It is simple to splice a cable with a simple knot as seen here:

https://sheldonbrown.com/heroic-repairs.html

The heroic Sheldon Brown is showing a cable joined with a square knot. I doubt it'll fit.

Can you at least explain what you’re trying to accomplish by soldering the cable?

The cable needs to be "grabbed" by the lever and advanced while one is pulling on the lever. I hoped joining the ends with solder would allow the grabbing action. But the knot of solder material always stopped the cable from advancing. And now looking at the torn-down apparatus I still don't see how this thing works.

Evidently you are doing it wrong, or the mechanism is broken. You should probably take hte advice of those who have replied, who know what they are doing.

Solder is not necessary. The new shift cable has a head that fits into the shifter, and this is what the shifter pulls or releases when shifting. With the tension released, you can just slide a new cable in until the head seats in the shifter. Have you watched any YouTube videos or the like?

Precisely. The HEAD is the thing that the shifter grabs. The blob of solder becomes a 'faux head' that will be interpreted as the signal to stop feeding in more cable.

Start with a new shift cable (different from a break cable)

more importantly either 1) take it to a shop 2) find a friend that know what they they are doing or 3) read in detail and follow various online how to's that have been offered as you are clearly in over your head

So I gave up and took it to the bike shop nearby. He said I had the wrong derailer for the front cog wheel. Charged $100 for looking at it and we left it because they can't work on in before the 19th. Then as I was leaving he ran out to tell me my wheel was bad. So $100 to look it over and replace the front derailleur and I'm guessing another $100 for a new wheel. When I bought this bike new from KMart in 1990 I paid $40.

I am repeating but for the shfiter look for a friction thumb shifter it is super easy
for the wheel what does bad mean? needs truing? bearings gone?

I would really look to get different bike than invest any more money in this bike. you could find something used for 200 to 300 that would be a ton better than this bike

You're right, of course. The truth of what you say is so self-evident, in fact, that I am convinced that no such transfer of wealth from the o.p. to an LBS in their state has taken place.

It was a 3-minute conversation. Later I realized I was in a similar situation last year when I wrecked my scooter. I took it to the store where I'd bought it and they said repairing would cost more than buying a new one. I opted to just abandon it there.

I'm guessing there are bulges in the wheel that would require a lot of time truing it back to normal so buying a new wheel would be easier.

By the time I get it back it will be snow season. Luckily, I have a trainer.