Personally, I prefer 'and the like' as this refers to similar time of vegetables (those that are alike) to the ones described. If you use 'etc' or 'and 'so on' you are potentially saying an never-ending list of vegetables. In which case you may as well say 'I like to eat all vegetables'. In saying 'and the like' you are restricting the range to just those vegetables similar to the ones you have described. Actually, we are more likely to say 'I like vegetables...' rather than 'eat' as it is generally taken for granted that you were talking about eating them. Alternatively, you could say 'I like growing vegetables...'

I agree with Powers. They're all very similar and most people would use them interchangeably. In this sentence, I personally would use "etc." but that's just my style. "Etc." means "and all the rest" which is very much like the other two options. The differences are very subtle.

Honestly, they can all be used. At least that is what I believe, but then again I am not an English teacher.