Learning niceties, like “Hello,” “Goodbye,” “Please,” “Thank You,” “Excuse me,” “Where is...”, etc. is alway helpful, but are you wanting to carry on involved conversations with locals? You may know that Rick Steves’ phrase book is a 3-in-1 French/German/Italian publication, so those last gauges would seem to be the “Big 3” worthy of a book by him. As you mentioned, speaking Spanish, being a Romance Language like French and Italian, would seem to be a helpful foothold (tongue hold?) for communicating any of those languages, but maybe not so much once you’re over there. It can depend on the situation, and with whom you’re speaking.
More and more over the past 20 years, Europeans, especially those in the Travel industry, speak at least some -or fluent- English. If you’re needing to buy something, or get directions, people in bigger cities or towns are likely going to be able to help you, even if you don’t exactly speak their native language. Smaller villages, and older (maybe 40+ years old) residents, who aren’t running a hotel, driving a taxi, selling tickets to a museum or transportation, or waiting a table at a restaurant, might not speak English (or even Spanish) so readily.
I’ve studied French, and can speak and read some Spanish. Unless trying to figure out a sign, where that’s helped some times, if you’re planning on talking with someone, then speaking, not reading, will do you more good. Italian in restaurants and catching trains has been where I’ve mostly wanted to use it. German - I can order 2 beers, and that’s about it, so that’s where I would need to start learning, but that’s just me.
Where are you spending the most time, and will you be in less-populated parts of countries? Or will you be in bigger tourist-orientated locations, where English speakers are pretty present?