As far as solder goes, you can only get the lead-free stuff now. In most cases that's good enough, but for older equipment you might have to recover the finer leaded stuff from other circuit boards you work on.
Apart from the items you've named already, I think the major item you're missing is a multimeter. You use it to test continuity and circuit paths, and you can also test voltage and resistance when diagnosing problems. I like the ones with the analog needles because you can see when signals rise and fall sharply, but digital ones are just fine too.