Good grief. It’s not your father’s Kester 44 anymore. I just found the remains of a spool of 60/40 that my father probably bought in the 50s, possibly even earlier.
As far as silver bearing solder goes, I have always used one of the Stay-Brite varieties.
I used to use an IsoTip rechargeable soldering iron when I was in the installation business in Alaska, but only a last resort when AC power was not available and a butane powered iron was overkill. Battery powered irons suck, even the best of them.
As far as the Weller goes, I just read the reviews on Amazon. Get rid of it.
Buy a commercial grade soldering iron or a soldering station. With a soldering station, look for plated replaceable tips, a burn proof flexible cord, and some form of automatic temperature control, not just a rheostat.
Look for the same things in a soldering iron, but be ready to give up the automatic temperature control, except for a few rather expensive models.
25-35 watts is fine for an iron that isn’t temperature controlled. You can go higher in a temperature controlled soldering station, as long as small enough tips are available for decoder work.
Even a professional grade 12 watt iron such as the Antex M12 will be fine for what your are doing, but it won’t be as versatile as the 25-35 watt model.
If it says the cord is silicone, you should be good to go in the burn proof flexible department.
My advice, buy the soldering station and be done with it.
I agree with CG and from my own experience, battery powered irons are a source of frustration and just not worth it. Get a good 25-35 watt iron and upgrade to a soldering station later.
And prior to the first use. The first time you plug in a new iron or after replacing the tip you should be prepared to coat it with solder and then wipe it off.
The best cleaner is the copper wool type, beats the damp sponge any day. My soldering sttion came with a holder that included one, but you can buy standalone iron holders (not a bad thing to have - beats laying the iron down on the bench and having it roll off the edge, or you whacking it with a hand or elbow) that have the copper pads in them. You can also usually find the pads in the grocery store, next to the steel scouring pads (NEVER use one of those on a soldering iron) - the soft copper ones are meant for scouring brass pots and pans - like the soft tip of a soldering iron, steel scouring pads will ruin the brass pans.
That iron will just eat batteries. AA batteries just can’t delivery the kind of power for soldering. Way back when, almost 30 years ago, we had a Wahl IsoTip cordless iron, which actually worked quite well for use out on the shop floor (lots of 480 volt outlets, not many 120 volt ones!) but it ran down fairly quickly. FOr major jobs we jsu tpulled the board and took it back to the shop and used a regualr plug in iron, but for here and there field work, it was fine. The difference being the Wahl uses nicad batteries which can delivery much more instantaneous power than alkaline batteries. And the good quality Wahl rechargeable iron costs as much or more than a decent budget soldering station. Also, a battery operated iron needs a different technique - you have to hold it OFF the joint, press the button, and wait the 15-20 seconds it takes to heat up, then touch the tip to the wire to be soldered. Compared to an already hot plugged in iron which you just pick up and apply to the joint. Seems like a minor difference but if you are used to a plug in iron and try to use battery iron the sme way, you WILL melt insulation or damage electronic components if you apply the tip to the joi