The literal and most common meaning of "suck" is to draw into the mouth by contracting the muscles of the lips and mouth to make a partial vacuum. This is the definition that applies in all of your examples.
"Suck" and "suck on" are pretty much interchangeable, and in your latter 2 examples, either could work:
Don't suck your thumb / Don't suck on your thumb.
Sucking a lollipop / sucking on a lollipop.
(although "sucking your thumb" is probably the most idiomatic in that example)
However, when it comes to a baby feeding, it is idiomatic to say "sucking at the breast", because it is as much a statement of their location and position as it is what they have in their mouth. It may even be considered an abbreviation of "suckling", which, as a verb, means to feed (a baby or young animal) from the breast or teat.
As an aside, the expression "suck at" something is also a modern colloquialism meaning that you are "bad at", or "not adept" at something.