Conversion from MTS to MP4 leaves low-quality or horizontal lines

[BRewind]

I recently purchased a Canon HF10 to start shooting video and putting things up on Vimeo for family. I am new to the world of video conversion and as you well know, the Canon saves in MTS format. I have tried several conversion programs, bitrate changes, etc., and my videos will either be converted to (1) a lower-quality, pixelated version of the video or (2) a good-quality video where horizontal lines show up along the edges of objects that are moving (this could be when the camera moves and the object is stationary or the object moves and the camera is stationary).

Are there any pointers on how to convert these videos without either of the two outcomes?

[Baldrick]

Sounds like your video is interlaced. Deinterlace or record in 24p(progressive) if possible.