Well, this isn't my top list of personal favorites, but for someone who isn't all that familiar with the genre, this is where I'd recommend starting. The first two choices are tied and the other three are kind of any given order.
Godzilla '54 - For people who are more into modern films, this movie is a bit of a hard ask. It's pacing is on the slower end and with the standard of current effects that we have today, this movie feels a bit quaint. However, those factors aside, this film stands out as a must see for so many reasons. Not only is it the original film that started the whole franchise, it was also a pioneer film in the realm of Tokusatsu special effects. It's a dark and serious film brimming with social commentary and heartfelt drama and in my opinion it is, and probably always will be, the best example showing that Kaiju films can be deep, artistic, endeavors.
Pacific Rim - On the other end of the spectrum is Pacific Rim. It embraces the more colorful, adventurous side of Kaiju films. It has creative monsters, flashy colors, and just enough storyline to move the film from one action scene to another. This film is all about the spectacle and action and is fun for the sake of fun.
Giant Monsters All Out Attack - Somehow, this movie went from being a film I was "meh" about to being in my top five favorite Godzilla films of all time. Story wise, it strikes a good balance between social commentary (if you're willing to swallow mysticism in story telling) and just a good, adventurous romp. At the same time, it's also one of the best examples of Tokusatsu in general and Suitmation in particular can be used for effective storytelling and spectacle. I've probably watched this movie half a dozen times in the past couple years, I enjoy it so much.
Cloverfield - It's a western kaiju film. It's a disaster film. It's a shaky cam, found footage film. It's got a little something for everyone. What I think this film has going for it that a lot of kaiju films sometimes struggle to do, is show the sheer scale and terror a kaiju brings from a human perspective. If you haven't seen it, Gareth Edward's Godzilla pulls this off by showing the monsters pretty much only from camera angles that would be possible by everyday people. However, I feel that Cloverfield's found footage technique does it better.
Any King Kong Film - Each one is a product of their era, for better or worse. If you want to get to know the character though, the story has been told many times before. The original '33 film sits on so many top lists, it's one of those movies that stands the test of time. That said, it's been so long since I've seen the original, there's probably non-PC elements in it that wouldn't get a pass today. Same probably goes for '76 as well. I actually really like Skull Island myself, if only because it tells a different story.
Heisei Gamera Trilogy - I will readily admit that I have yet to see this trilogy, but every person I've ever talked to who has says it does not disappoint. It not only redeems Gamera from his schlocky reputation, I've been told that in story, special effects, and monster design, it surpasses a lot of the Heisei era Godzilla films. I look forward to adding it to my collection.