What films to watch before Avengers: Infinity War for better understanding?

We don't really know, since all we have to go on is the trailer. This is the culmination of the entire Marvel Cinematic Universe to this point, and every superhero we know of is going to be in it. So, the easiest answer is all of them.

Just based on what happens in the trailer, though, we can at least say the following (NOTE: These are not in order; after deciding what you're going to watch you should watch the movies in the order they were released):

Mandatory

• Avengers and Age of Ultron - Infinity War is a direct sequel to Age of Ultron, and in those movies, we learn about Thanos, the Tesseract, and the Infinity Stones, plus are introduced to the Avengers roster itself.
• Captain America: Civil War - This is practically an Avengers movie as it is; Infinity War seems to pick up the story of the "fugitive" Avengers where it ends in Civil War, especially when it comes to Steve Rogers.
• Thor: Ragnarok - Chronologically, it seems Infinity War begins very soon after Ragnarok ends, and picks up Thor and Loki's story from that point. From the trailers, we can guess that events in Ragnarok play a crucial role in kicking off Infinity Wars.
• Black Panther: Not out yet, but the trailer implies that T'Challa and his home country of Wakanda play a major role in Infinity War, so you'll probably want to see that too.
• Guardians of the Galaxy: A good chunk of the movie follows the Guardians, and one in particular has a very close relationship to the film's antagonist. This film is also the first to actually explain what an Infinity Stone is, though you'll get a refresher in Infinity War.

Recommended

• Doctor Strange: Also introduces us to an Infinity Stone, and some other non-Avengers characters.

• Thor: The Dark World: The last of the known Infinity Stones, unfortunately, is in this movie. Though, if you want to skip it, all you need to know is:

  The Aether is an Infinity Stone, Jane Foster found it, and after Thor beat the bad guys, the Asgardians gave it to The Collector for safe keeping.

• Guardians of the Galaxy: Volume 2: The roster changes a bit from the first movie to the one we see in Infinity War, though you really just need to know:

  Mantis has psychic powers.

Suggested

• Ant-Man, Spider-Man: Homecoming: Introduce characters that haven't been in an Avengers movie yet. They do appear in Civil War, though, and it's unlikely their solo backstories will play too heavily into Infinity War, at the least due to time constraints, so you may be able to give these a pass.

The rest of the movies (Iron Man series, Captain America series, Thor, and especially The Incredible Hulk) you probably don't need to see if you're short on time. You can pick up the majority of the characters background and abilities from the Avengers movie, at least, likely enough to get you through Infinity War.

Can anybody tell me what moves I should watch before Avengers: Infinity War for better understanding?

According to Kevin Feige, no additional Marvel films need to be seen in order to understand what's going on in Avengers: Infinity War.

QUESTION: Can you enjoy the film if you aren't up to speed on the Marvel Studios movies to date?

KEVIN: We believe that every movie we make needs to feel like a standalone piece of entertainment -- that you can walk in cold and enjoy, with a beginning, middle and end. We always ask that question in our research screenings. Do you feel you needed to have seen the other movies?

People who have not seen the other movies say 'No.' People who have seen the other movies can see the connections. That's the balance that we want to find: making a film that will appeal to people who've never seen one of our movies. It's an entertaining self-contained story, but it will work on a much deeper level for people who are following along.

This in mind, if you want to know why some characters look/behave they way they do, and the dynamics of the relationships between some characters, then yeah, watching a few of the previous films would be a really good idea (in which case I would recommend something extremely similar to @KutuluMike's list).

However, do any of the previous Marvel films actually need to be watched, as perhaps a typical sequel might demand of its prequel(s)? Not at all.

Reviews that I've read of Infinity War do suggest that the movie is enjoyed more by people that are more invested in the characters and that as a stand-alone movie it doesn't work as well as Keven Feige suggests.

The other suggested list answer is excellent but with full knowledge of Infinity War I do slightly disagree with its suggestions.

I've attempted to cut down the list as small as possible for people that don't want to spend forever filling in the blanks in their MCU coverage. I don't think its critically important to explain the location and capabilities of all the Infinity Stones prior to Infinity War, which explains why I skip the underwhelming Thor The Dark World. As an example, by missing this movie its not clear why the Reality Stone is with The Collector, but at least that character and his interest in the stones is clear from Guardians of the Galaxy. I feel this is justified as the Aether from this movie doesn't really act like the Reality Stone.

This is also not a "best of the MCU". If that were the case, I'd probably drop Age of Ultron, elevate Black Panther and include Captain America The Winter Soldier. It really is an attempt to introduce characters and concepts in as few movies as possible.

Mandatory

• The Avengers. This movie can substitute for the origin story for much of the core team allowing you to skip all the other "phase one" movies. They do a great job of subtly filling you in, for example it introduces what happened to Cap in his first movie and explains how he ends up in the present day, and who the Asgardians are.

• Guardians of the Galaxy. This is the movie that properly introduces the Infinity Stone concept to the franchise. The Guardians are very prominent in Infinity War, especially as two characters have an important relationship to Thanos, the primary antagonist.

• Avengers Age of Ultron. Probably the weakest movie on the mandatory list and I have toyed with dropping it down to being recommended only. Importantly it introduces Tony Stark's fear of failure and planning for the future "big bad" that will threaten the earth. Its also the origin story for Vision and why he has the Mind Stone.

• Captain America Civil War. This is an Avenger's movie in all but name and will explain the composition of the teams in Infinity War and introduces Spiderman to the MCU. Peter Parker's relationship with Tony Stark is very important and, although this is expanded upon in Spiderman Homecoming, you see enough of it here to understand how Tony feel protective towards his protegé. It also acts somewhat as an origin story for The Winter Soldier and gives a glimpse into the world of Black Panther and Wakanda. (It also includes enough of Vision for you to understand who he is if you happened to skip Age of Ultron)

• Doctor Strange. Doctor Strange plays an significant role in Infinity War so its important to understand him and his capabilities. Except for a cameo in Thor Ragnarok you don't see him in any of the crossover movies to date.

• Thor Ragnarok. The events of Infinity War happen shortly after the events of Ragnarok. This movie also allows you to catch up with the character arc of Loki and explains much of the motivation for Thor in Infinity War.

Recommended

• Guardians of the Galaxy Vol 2. Along with the Peter Parker - Tony Stark relationship, the Guardians - Thanos relationship forms much of the emotional weight of Infinity War, so watching this movie is also recommended. It also introduces Mantis to the team.

• Black Panther. Although plot-wise this contains very little to link it with the overall MCU story arc, Wakanda is a key setting for Infinity War and its a great movie to round out the experience if you have time.