Seeking advice from the forum

There's a few with some tight continuity. As far as what Hulu has, here's what needs to be viewed in order:

The Heisei Series

These movies try to have strict continuity, despite how much they flub it up and don't use it. But for the sake of consistency it's best to watch them in order. It's important to note that the first two films made for this continuity are not on Hulu (The Return of Godzilla & Godzilla vs. Biollante). So you'll have to go in to it with the understanding that a new Godzilla has appeared twice since 1984--And before that only the events of the original 1954 film had taken place:

Godzilla vs. King Ghidorah
Godzilla vs. Mothra
Godzilla vs. MechaGodzilla II
Godzilla vs. SpaceGodzilla
Godzilla vs. Destoroyah

The Millennium Series

The Millennium movies are an anthology series with no continuity between them. With the exception of two films, the only connection they generally have is to the 1954 original. Any important information from the original film is usually recapped in some way. You can watch any of the following in any order you wish. But this is the order they were released:

Godzilla 2000
Godzilla vs. Megaguirus
Godzilla, Mothra, King Ghidorah: Giant Monsters All-Out Attack
Godzilla: Final Wars

The Kiryu Saga

The two Millennium movies I spoke of that were the exception are usually dubbed "The Kiryu Saga" or "Kiryu Duology" by fans. Like the other Millennium movies, they're sequels to the original Godzilla and some other Toho kaiju flicks. Again, pertinent information is recapped via flashbacks:

Godzilla against MechaGodzilla
Godzilla: Tokyo SOS

I generally consider Godzilla vs. SpaceGodzilla the worst of the entire series and Godzilla vs. Megaguirus and Godzilla Final Wars aren't too far behind. I know many who would disagree, but I come at the series from a very different perspective than most. The 1998 movie is a borderline taboo subject among the fandom, so we usually just ignore it.

But, as with anything, try them all out. Tastes vary and I have no idea what you might be looking for in these flicks.

G. H. (Gman) Appreciated- thank you. Lots of watching to do. I just want to watch with as much continuity as possible. 

Hope you enjoy them--Keep us posted about your journey through those films. I sent you a PM with an aid on said journey.

Just got through with 2014. Pretty good. I felt like I did as a kid- the plot and story was pretty good. I ultimately didn't care about the people and was always wondering if Godzilla was ok after a fight. I could sympathize with the male and female monsters since they were just doing what they did and humans just got in their way. I would welcome more Zilla and monster time though. Even in Gojira, it felt excruciatingly long to get to the meat and potatoes. Maybe that is just how the stories are written- I am from USA and demand immediate gratification LOL! Over all, I enjoyed it very much.

One thing I disliked and happens with other movies was Soldiers using small arm weapons against gigantic creatures. I guess audiences need to see and hear rat tat tat. It is ridiculous and an insult to the military in general. At least with 1998 they used believable weapons from fighter jets to kill Godzilla.

Can't really add my informationwise, but just opinion based unfortunately for me that section has a lot of dubs, but a few greats.

If you start to go through this and decide the start skipping them the first one I'd mention is Godzilla vs Megaguirus. Depending on the tones you like Final Wars may be another one to miss out on.

^Durp brings up an interesting point. A lot of us around here watch the Japanese language version not just to avoid the hideous dubs, but the massive inaccuracies in them.

I've avoided the Millennium Series dubs, but I know the same company that dubbed them also did the English dubs for the Heisei series and they were absolutely atrocious.

Meanwhile the re-writing in the English dub scripts for both Godzilla vs. Destoroyah and Godzilla, Mothra, King Ghidorah: Giant Monsters All-Out Attack are among the worse. A lot is lost in translation and context in those two.

I imagine Hulu is only showing the dubs.