Ordinary oldschool pine rosin for soldering should be a good flux for beginner. It does not require cleaning after use and can indicate overheat. But do not put too much rosin in the place of soldering, otherwise the quality of the soldering will be poorly seen and it will be necessary to scrape the rosin out to check the quality.
Solder? A simple 61% tin-lead solder (like Russian POS-61/ПОС-61) is a good choice. It is better to buy solder in reels. Diameter of the solder should be approx. 1mm. The presence of rosin (flux) inside the solder will not be bad, it simplifies the soldering and saves the external rosin.
The soldering-iron should be approx. 40 watt. 25 watt models are too cold for RoHS (lead-free) solders that are used in PC hardware since late-1990s, and 60W soldering-irons are too hot for motherboards. I am using a old soviet 60 W soldering iron to replace capacitors in modern boards, but there it is necessary to solder very quickly and precisely, for a beginner such a soldering iron will be inconvenient. It's more like a thermorektal cryptanalyzer from communism era, not a soldering iron. 40W are very easier to use, but they're working some slower than hot 60 watt models.
edit: fix typo
2×Soviet ZX-Speccy, 1×MacIIsi, 1×086, 1×286, 2×386DX, 1×386SX, 2×486, 1×P54C, 7×P55C, 6×Slot1, 4×S370, 1×SlotA, 2×S462, ∞×Modern.