I wanted to do very crisp (maximum resolutions/MB) videos that uses a style similar to Khan Academy (blackboard style with several stroke colors throughout the video, maybe a few still pictures here and there) but needs to be lightweight in size. What's the best format to save these videos and how should they be compressed?
Format doesn't really matter, I usually use something fairly common like .mov or .mp4. The more important thing is the codec. Usually when exporting for online upload, H.264 is the most common codec because of its balance of size and quality. For resolution, either 1080p or 720p. Most people can't tell the difference between the two, but 1080p is the standard so I generally stick with that.
Another thing to keep in consideration is color bit-depth. I'm not sure if the H.264 codec or different programs allow you to export with different bit-depths, someone else can comment on that (or downvote, whatever). Assuming you can control bit-depth, I would go with 10. Most consumer grade cameras only shoot with 8 bit-depth, but you can get higher with the animation depending on what you generate it in and how you render. The extra bit-depth will improve the overall crispness.
One last thing to keep in mind is how you render the animations. I know you probably intend to create 2D, but I'll describe my process with Maya which is 3D, and hope all the information pretty much transfers over. First, export at slightly higher than 1080p. I'd go with something like 2K. The downscaling will improve your quality and it's just nice in general knowing you have that extra space to work with. Don't enable motion blur, or at least keep it to a minimum. Other than that, just experiment with the render settings to get the effect you want.
As an example, this Khan Academy video is uploaded as 720p to YouTube.
Youtube's upload recommendations for 720p are:
Video Bit RateRate Stereo Audio Bit Rate
Mono Audio Bit Rate: 128 kbps
Stereo Audio bit rate: 384 kbps
High quality: 720p 30,000 kbps 128 kbps 384 kbps 512 kbps
resolution: 720 p: 1280x720