You bought a reasonable solder. Sn63Pb37 is a (close to) eutectic alloy, so it has a bit more of the more expensive element (tin) and is a bit better than Sn60Pb40 solder. As a result, it melts at a bit lower temperature and (more importantly) it's a bit easier to use because it solidifies suddenly rather than going through a mushy phase where it's easy to create a "cold" solder joint if the parts move during cooling.
0.025" (0.635mm) is a reasonable size for general purpose. I use 0.8mm and 0.38mm.
Personally, I prefer Kester 44 because the "no clean" flux leaves a nasty residue that is very difficult to clean off. For most applications it's just fine, but for sensitive high-impedance analog circuitry it can be a problem. I had an application with 5-50K resistors where it was a problem, because of extreme accuracy requirements. The contract assembler had used no-clean, contrary to instructions. The Rosin RMA flux may look ugly if you don't clean it, but it seems pretty inert, and it's easy to clean with solvents. I doubt you'll see any problems with the one you chose.
You say "solder station". It's temperature controlled? That helps.
You don't need to buy any extra flux for through-hole parts. You might want to buy a flux pen (eg. Kester #186) if you intend soldering surface mount parts- it helps a bit. They look like a felt-tip marker but dispense flux. Unfortunately, they seem to be a bit hard to come buy these days. Bulk flux attracts Hazmat shipping charges.
Make sure you're dealing with "fresh" parts. If they've been laying around in some surplus shop oxidizing for a decade or two, they're not going to be all that easy to solder!
If the tip is well wetted, you should be able to touch it to the two parts at once, feed the solder into that pool, and stop when you have a fully wetted joint. It should be shiny and smooth when it's done. Practice makes perfect.
I suspect if you've had problems with soldering, other than technique, your parts are the source of the difficulty rather than the solder or soldering iron.
Your problem is almost certainly not a direct issue with the solder itself but with the flux in the solder. Many fluxes leave a conductive film on the board which must be removed for circuits such as yours.
Note that RTC chips try to be extremely low power so as to preserve the battery as much as possible. As such, the biasing of the 32.768KHz crystal for the oscillator is designed to be rather low current (i.e. high impedance circuit). The flux conductivity can sometimes be low enough to completely swamp out and literally short out the oscillator preventing it from working.