Adrian’s Digital Basement published a nice video where he repairs a C64 with only low-cost equipment. The video is on the long side (as always), but it’s quite educational and gives some good basic diagnosis and soldering advice for the novice.
He uses a cheap Hantek oscilloscope ($60, it’s not great but does the job, much preferable over a logic probe), a Pinecil soldering iron ($25, and it’s actually good), a cheap multimeter ($25), and a soldering pump and braid. The total equipment cost is only a little over $100, maybe $150 if you include supplies.
Had this been available back in 2014, I might’ve had quite a different soldering setup :). I’m happy with my Weller soldering station ($200) and Rigol oscilloscope ($500), they’re quality products, but they cost a pretty penny and I could’ve saved a lot of money for the modest amount of electronics work that I do. And I don’t even have a desoldering gun or hot air station. Luckily my multimeter is a cheap one :).