Tell a story with your screenshots. I would even recommend including a few words with each screen that highlights the value your product provides. Make sure there is a flow to the screenshots that tells your overall narrative.
GIFs can tell a powerful story in a few seconds and I would highly recommend using at least one of them. We took many hours to create impactful GIFs and I think all the work really paid off.
A couple tips:
Don't have too much movement. Because GIFs loop automatically, if things are moving/zooming all over the place it can be overwhelming and jarring. Focus the action on a few key interactions. Keep the frame focused on a few key interactions.
Fake it if you need to. Sometimes there are pieces of your app that don't lend themselves to helping communicate in a GIF. Use video editing software to skip steps, blank things out, add things in, and do what it takes to communicate quickly and effectively. The point is not to decieve (please don't mislead people) but to quickly communicate what your product is all about.
Use a wide aspect ratio if you plan to have multiple GIFs next to each other (ours were 1000 x 600 or 5:3). If your GIFs are too narrow, you will end up with GIFs playing next to each other and competing for the user's attention.
The thumbnail is the "icon" that shows on the homepage. It can be a GIF. Be careful though. It is really small so if you don't do it right, your GIF may just feel cheesy or distracting. If you can pull it off in a way that actually compliments or tells your story in a tasteful way, go for it. Bottom line: using a GIF thumbnail will tend to make your product stand out but not always in a good way to be careful.
A lot of the most successful Product Hunt products use a video. However, videos take a ton of time and/or money to do well and they ALWAYS show first so if a video isn't your best foot forward, don't waste your time. If a GIF can communicate just as well, I would recommend that — they are easier to pull off well. If, however, you have the means to create an amazing video, it can be a powerful asset to draw people into your story.