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Which languages should one be familiar with if they wanted to get into historical linguistics? Specifically, Indo-European linguistics, reconstructing Proto-Indo-European etc. Which ones would be most useful and typologically interesting?

Knowledge of language families is needed. For a start, it should be very helpful to know one language from the same family as your mother tongue, and at least practical experience with one outside that family, but it's not really required for the theory. Rather, the theory should inform which language you would want to learn. But you cannot feasably learn enough languages, so the choice needs to be economic. That only you can decide in the end. Don't be mistaken, even Greek,e.g. is a language family (thus called Hellenic). You might want to reach a basic level in many, and specialize in few.

One reason to learn a language is to read relevant literature on the topic. German, French and Russian should suffice except in the case of a sub-specialization in South Slavic, for example. Then there are languages that you need to learn as objects of research, where ideally you would learn all of the old languages from each branch of IE. Ancient Greek, Latin, Sanskrit and Hittite are the most obvious, then Old Irish, Old Norse, Old Church Slavic... The list goes on. I don't mean that you have to gain mastery of each of the languages. It also depends on why you want to reconstruct Proto Indo European, since that has kind of already been done. Do you have a more nuanced interest?

Depends how much effort you want to put in, and you should also consider what you actually want to find out.

Minimum effort, just to pass a course: Check the course material and learn some basics about the languages mentioned there. It'll be more than the others will do, you'll get a good grade and have a good basis for a thesis or such.

Minimum effort, to satisfy personal curiosity or to work in the area: Study what you are curious about, you will automatically stumble upon languages that might answer some questions, and you aquire as much about them as needed. Depending how much you have to learn, you may want to branch out a little to make the best use of the effort: Translations, research by other people and so on.

Medium effort: You know English. As it's a mix of Germanic and Romanic, you may want to study a more Germanic and a more Romanic language to complete your knowledge in the area, and then one language that interests you personally to get a different perspective. Besides your linguistic reasons I suggest also adding practical uses in the choice of languages - opportunity to earn money for your studying and travels, for instance. Not a too crowded field, especially in the low-income area. Other languages you just learn the basics.

High effort: Try to aquire 5 or 6 completely different languages and learn to communicate somewhat in several other languages and regional dialects. As far apart geographically as possible, and with travels to become fluid and find out about actual usage and fine details. In your case, that means a Germanic and a Romanic language, Russian or something related from around Russia, at least 2 languages from India, and also at least to beginners level 2 completely different languages - something African or native American and something east Asian, so you have a comparison. A conlang or two (especially the ones based on research about Indo-European languages) can also help.

If you want to bring humanity forward, try to not just delve into the past, but also give speculations about the future a scientific footing. How languages managed to improve clarity over time or compared to others. Or be more concise without loosing in other areas. Easy to learn. Understandable for someone who is not part of the culture (independence from regional context). Useful to express complex ideas (math, physics, complicated machines...) in understandable ways (ideally with tests) even for listeners who have no idea about the subject. Useful to differentiate between fine details without making the listener or reader miss them easily. Easy to understand acoustically, even in noisy conditions (phonetic separation of digits and numbers, for instance). Easy to pronounce for locals or for learners from other regions. Logical. And/or some other things I probably missed.

None. Personal competency in languages is not what it means to be a linguist. While you need English as it's the common language of scholarship (though of course much research is published in other languages; if you wanted to do historical research in China, learning Chinese would be essential), in general linguists analyse languages without needing to be fluent in them. Even if you do primary description of minority languages you don't need to know them fluently, though of course it helps.