As for me, both are correct.

What do you suggest I do? is correct. However, I recommend saying, "What do you suggest?" I hope this helps!

I used to have this doubt too. Both are correct, but the second sentence is little wrong, the correct way is ''What do you suggest FOR me to do?''. Anyway, you can use both sentences in your daily :)

1 is correct. 2 would be correct by adding "for": What do you suggest for me to do?

1 is correct. You could also say "What do you suggest that I do?" 2 sounds like it's correct, but it's unnatural sounding. So... it's incorrect.