The very first mission I’d take in time travel would be to go to the Library at Alexandria the night before the first major fire.
This would actually take at least two trips — one to find out where and how extensive the damage was going to be, and a second to the night before in order to rescue all of those manuscripts that would be lost and substitute for them adequate fakes to use as just so much fuel.
My purpose would be purely in the interest of salvaging this remarkable written legacy. Some people might try to “help” things along by preventing the fire (assuming causality would permit it), or by distributing this knowledge around so that the “Dark Ages” would happen. Meddling like that is so dangerous that I’ll skip the obvious discussion.
I think the only useful thing to do is to rescue these long-lost works. Personally, I’d love to have a copy of all of the works by Psappho. She was so highly praised by her contemporaries (names familiar even to us) that it’s a tragedy that almost all of her work is unknown to us. I’d also love to see the writings of Archemedes. It’s clear that he stands out as a genius not only of his time, but of ours as well.
There are so many works like these that have disappeared. I can’t help but be curious about how much knowledge was in the hands of western man more than two millennia ago. How much did we lose? How far behind did we fall — or, to put it another way, how long did it take for us to recover?
The Library. First trip. Definitely.