Which Star Wars movies are required to understand characters and plot elements of Solo: A Star Wars Story?

I haven't seen it but looper explained it pretty well:

Solo's story, however, is a done-in-one. You won't find any cliffhangers here, and you don't need to watch any previous movies to understand what's going on. The Last Jedi this ain't. You'll probably get more out of Solo if you have a passing familiarity with the characters — one joke in the second Solo trailer, for example, only lands if you're up on your Star Wars catchphrases — but it's not necessary. If you're not an expert on everything that happened a long time ago in a galaxy far, far away, you should be fine.

Also from Wikipedia:

A stand-alone installment set between the events of Revenge of the Sith and A New Hope, it explores the adventures of younger versions of Han Solo and Chewbacca, who also meet a young Lando Calrissian.

I watched the film last night. I can confirm that you don't need to see any of the other films to enjoy this one. That being said, there are several references to the original trilogy that will escape a Star Wars newbie (I think most of them will not be noticeable, either - all but a few are fairly subtle).

Although I agree with others, despite that Solo: A Star Wars Story tends to inform The Original Trilogy (A New Hope, The Empire Strikes Back, and The Return of the Jedi) and helps to full circle Han's life (The Force Awakens), one does not need to see those films or other materials to enjoy 'Solo', but one will definitely get more out of it, and may be surprised or delighted with the plot twist ending (The Phantom Menace, Clone Wars, Star Wars: Rebels), if they had.

Why it might still be better to watch most of The Skywalker Saga films (Episodes I-VII), especially The Original Trilogy, before seeing Solo: A Star Wars Story:

1. By showcasing younger versions of established classic characters, one will not only see how Lando, Chewy, and Han meet, but you will also see riffs or juxtapositions of scenes from the original trilogy with these characters.
2. In similar fashion there are new characters such as Beckett and Qi'ra that inform whom Han Solo will become in the original trilogy, as Beckett has characteristics of future Han and Qi'ra tends to juxtapose Han's future relationship with Princess/General Leia.
3. As mentioned in the paragraph above, this story doesn't just inform the original trilogy by explaining things mentioned in the original trilogy, or with callbacks, but Han in this film is also not quite the Han many know from that era, being a bit more optimistic ("I have good Feeling about this!" vs "I have a bad feeling about this"), putting him closer to the Han Solo presented in The Force Awakens, who comes to embrace notions of The Force, where previously he was dismissive or doubtful. It helps full circle his character's full arc (thus far).
4. The ending reveals who's the head of Crimson Dawn. For some this ending will be super shocking, because for those that only watch the films (and no other media), would have thought this character to be dead (The Phantom Menace), but those that do follow other media (TV Series: Clone Wars, Rebels), might be delighted, because to them, it would make perfect sense that this character would be here. Without knowing either, this ending may not mean too much.

Solo is an origin story. It might actually help someone who hasn't seen any of the series to see it first. Then they would have a better understanding of why Han is the way he is. Now in my opinion everyone should see all of them. Yes, episodes 1 through 3 (the 4th-6th movies released, weird Anakin age jump from child to adult sleeping with the Queen, so awkward) were terrible, but it's all part of the greater picture. So yeah you can see Solo without seeing the rest.. it's not technically part of the saga, but why deprive yourself of such an iconic film series?