If it's on any HD screen you might want it at 1280x720p but that won't give the best quality on 1080p screens (nor UHD screens). I'm not sure whether every HD screen is compatible with 24 fps so you may want 1280x720p60 as that's a TV standard introduced when HD arrived in the US (as well as 1920x1080i30 - you could also try that). If 24 fps is compatible with most you might want to use that (though I don't think it is). US and other region TVs should be compatible with 1280x720p60 (even though European and Australian TV is normally 25 fps/50Hz). I'd probably do a version at 1920x1080 as well though as that's what most currently on sale HD screens probably are though (with the UHD ones being 3840x2160). Animate your artwork setting keyframes or whatever (you are best looking at After Effects beginners guides for things like importing files, moving things with keyframes). But for moving a layer you basically create a keyframe with the layer (eg. image) positioned at one position (x,y or for 3d, x,y,z) and at a different point in the timeline you can reposition it and create a new keyframe with the layer (eg. image) in a different position. After Effects will interpolate the positions inbetween (using whatever interpolation method you have in your preferences). A .mp4 will be better, with a codec like H264 for the final rendered video because it's likely to be most compatible with TVs/players I think. Basically, after you've finished any animation in After Effects, you can select "Add to Media Encoder Queue" if you want to render it to the H264 codec. In Media Encoder make sure the settings are what you want in terms of bitrate etc. render quality (eg. "Use maximum render quality" checked), audio etc. and click the green triangular button to "Start Queue". If the HDTV is old and it doesn't work with H264 you could try mpeg2 (but that will be either lower quality than h264 or need a higher bitrate to match the quality). I'm assuming your going to be playing from a memory stick and not writing to a disc for this (if you are writing to a Blu-ray disc it would still likely be best as H264 either 1280x720p60 or 1920x1080i30 - for USA at least for the best compatibility - written with Blu-ray authoring software. -- My recommendation though is to go through the After Effects beginner's tutorials first though so you know how After Effects works in terms of loading, keyframing, previewing, rendering etc.
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