I did some color correction and also some 60i to 24p conversion and I can see blocks/lines in the footage now. You can only notice if you look real hard. Why is that? This happens with various programs. Is it how the softwares process the footage? Anyone know what I mean?

It might be artifacting related to chroma subsampling. Try using uncompressed intermediates.

Chroma blur may help. Perhaps more so than using uncompressed intermediates (although it depends what codecs you're using to generate the "uncompressed" intermediate, and whether it's no codec or a 4:2:2 codec).

What program are you using?

2- Can you post a picture? Maybe you are talking about something else like interlacing artifacts.